diff --git a/ProbabilityParadox.ipynb b/ProbabilityParadox.ipynb
index 074fd2c..ae643a6 100644
--- a/ProbabilityParadox.ipynb
+++ b/ProbabilityParadox.ipynb
@@ -4,7 +4,6 @@
"cell_type": "markdown",
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- "deletable": true,
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@@ -20,11 +19,10 @@
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{
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- "execution_count": null,
+ "execution_count": 1,
"metadata": {
"button": false,
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- "deletable": true,
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"run_control": {
"read_only": false
@@ -84,7 +82,6 @@
"cell_type": "markdown",
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"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -121,7 +118,6 @@
"cell_type": "markdown",
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- "deletable": true,
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"run_control": {
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@@ -135,11 +131,10 @@
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{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 2,
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- "deletable": true,
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@@ -154,8 +149,6 @@
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- "collapsed": false,
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@@ -167,11 +160,9 @@
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{
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+ "execution_count": 3,
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"run_control": {
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@@ -188,7 +179,6 @@
"cell_type": "markdown",
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@@ -200,17 +190,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 4,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(1, 2)"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(two_boys, such_that(older_is_a_boy, S))"
]
@@ -219,7 +218,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
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@@ -235,11 +233,9 @@
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{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 5,
"metadata": {
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- "collapsed": false,
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"new_sheet": false,
"run_control": {
"read_only": false
@@ -252,17 +248,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 6,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(1, 3)"
+ ]
+ },
+ "execution_count": 6,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(two_boys, such_that(at_least_one_boy, S))"
]
@@ -271,7 +276,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -283,17 +287,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 7,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "{'BB', 'BG', 'GB'}"
+ ]
+ },
+ "execution_count": 7,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"such_that(at_least_one_boy, S)"
]
@@ -302,8 +315,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
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@@ -335,11 +346,9 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 8,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -357,7 +366,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -369,17 +377,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 9,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "{'BB/?b', 'BB/b?', 'BG/b?', 'GB/?b'}"
+ ]
+ },
+ "execution_count": 9,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"def observed_boy(outcome): return 'b' in outcome\n",
"\n",
@@ -390,7 +407,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
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@@ -402,17 +418,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 10,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(1, 2)"
+ ]
+ },
+ "execution_count": 10,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(two_boys, such_that(observed_boy, S2b))"
]
@@ -421,7 +446,6 @@
"cell_type": "markdown",
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- "deletable": true,
"new_sheet": false,
"run_control": {
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@@ -454,17 +478,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 11,
"metadata": {
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- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "196"
+ ]
+ },
+ "execution_count": 11,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"sexesdays = cross('BG', '1234567')\n",
"S3 = cross(sexesdays, sexesdays)\n",
@@ -482,11 +515,20 @@
},
{
"cell_type": "code",
- "execution_count": null,
- "metadata": {
- "collapsed": false
- },
- "outputs": [],
+ "execution_count": 12,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "['G2G3', 'G6B7', 'B1G7', 'B4B7', 'B4G6', 'G5G5', 'B5G4', 'G4B4']"
+ ]
+ },
+ "execution_count": 12,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"import random\n",
"random.sample(S3, 8)"
@@ -496,7 +538,6 @@
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@@ -508,34 +549,52 @@
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+ "execution_count": 13,
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- "collapsed": false,
- "deletable": true,
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"run_control": {
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}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(3, 4)"
+ ]
+ },
+ "execution_count": 13,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(at_least_one_boy, S3)"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 14,
"metadata": {
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- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(3, 4)"
+ ]
+ },
+ "execution_count": 14,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(at_least_one_boy, S)"
]
@@ -544,7 +603,6 @@
"cell_type": "markdown",
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- "deletable": true,
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@@ -556,34 +614,52 @@
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{
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- "execution_count": null,
+ "execution_count": 15,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(1, 4)"
+ ]
+ },
+ "execution_count": 15,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(two_boys, S3)"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 16,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(1, 4)"
+ ]
+ },
+ "execution_count": 16,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(two_boys, S)"
]
@@ -592,7 +668,6 @@
"cell_type": "markdown",
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"run_control": {
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@@ -604,34 +679,52 @@
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{
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- "execution_count": null,
+ "execution_count": 17,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(1, 3)"
+ ]
+ },
+ "execution_count": 17,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(two_boys, such_that(at_least_one_boy, S3))"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 18,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(1, 3)"
+ ]
+ },
+ "execution_count": 18,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(two_boys, such_that(at_least_one_boy, S))"
]
@@ -640,7 +733,6 @@
"cell_type": "markdown",
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- "deletable": true,
"new_sheet": false,
"run_control": {
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@@ -652,11 +744,9 @@
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{
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- "execution_count": null,
+ "execution_count": 19,
"metadata": {
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- "collapsed": false,
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@@ -671,7 +761,6 @@
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- "deletable": true,
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"run_control": {
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@@ -683,17 +772,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 20,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(13, 27)"
+ ]
+ },
+ "execution_count": 20,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(two_boys, such_that(at_least_one_boy_tues, S3))"
]
@@ -702,7 +800,6 @@
"cell_type": "markdown",
"metadata": {
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- "deletable": true,
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@@ -720,7 +817,6 @@
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@@ -734,11 +830,9 @@
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{
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- "execution_count": null,
+ "execution_count": 21,
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"run_control": {
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@@ -755,17 +849,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 22,
"metadata": {
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- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "['G4G5/??g5', 'B1B5/b1??', 'G6G4/g6??', 'B1G2/??g2', 'G5G6/g5??']"
+ ]
+ },
+ "execution_count": 22,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"random.sample(S3b, 5)"
]
@@ -774,7 +877,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
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@@ -786,17 +888,26 @@
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{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 23,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(1, 2)"
+ ]
+ },
+ "execution_count": 23,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(two_boys, such_that(observed_boy_tues, S3b))"
]
@@ -805,7 +916,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -823,11 +933,9 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 24,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -868,7 +976,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
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@@ -880,17 +987,29 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 25,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "P(two_boys | older_is_a_boy) = 1/2"
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "execution_count": 25,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"# Child Problem 1\n",
"Pgrid(S, 1, two_boys, older_is_a_boy)"
@@ -900,7 +1019,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
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@@ -912,17 +1030,29 @@
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{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 26,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "P(two_boys | at_least_one_boy) = 1/3"
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "execution_count": 26,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"# Child Problem 2\n",
"Pgrid(S, 1, two_boys, at_least_one_boy)"
@@ -932,7 +1062,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
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"run_control": {
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@@ -944,17 +1073,29 @@
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{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 27,
"metadata": {
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- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "| B1B1 | B1B2 | B1B3 | B1B4 | B1B5 | B1B6 | B1B7 | B1G1 | B1G2 | B1G3 | B1G4 | B1G5 | B1G6 | B1G7 |
| B2B1 | B2B2 | B2B3 | B2B4 | B2B5 | B2B6 | B2B7 | B2G1 | B2G2 | B2G3 | B2G4 | B2G5 | B2G6 | B2G7 |
| B3B1 | B3B2 | B3B3 | B3B4 | B3B5 | B3B6 | B3B7 | B3G1 | B3G2 | B3G3 | B3G4 | B3G5 | B3G6 | B3G7 |
| B4B1 | B4B2 | B4B3 | B4B4 | B4B5 | B4B6 | B4B7 | B4G1 | B4G2 | B4G3 | B4G4 | B4G5 | B4G6 | B4G7 |
| B5B1 | B5B2 | B5B3 | B5B4 | B5B5 | B5B6 | B5B7 | B5G1 | B5G2 | B5G3 | B5G4 | B5G5 | B5G6 | B5G7 |
| B6B1 | B6B2 | B6B3 | B6B4 | B6B5 | B6B6 | B6B7 | B6G1 | B6G2 | B6G3 | B6G4 | B6G5 | B6G6 | B6G7 |
| B7B1 | B7B2 | B7B3 | B7B4 | B7B5 | B7B6 | B7B7 | B7G1 | B7G2 | B7G3 | B7G4 | B7G5 | B7G6 | B7G7 |
| G1B1 | G1B2 | G1B3 | G1B4 | G1B5 | G1B6 | G1B7 | G1G1 | G1G2 | G1G3 | G1G4 | G1G5 | G1G6 | G1G7 |
| G2B1 | G2B2 | G2B3 | G2B4 | G2B5 | G2B6 | G2B7 | G2G1 | G2G2 | G2G3 | G2G4 | G2G5 | G2G6 | G2G7 |
| G3B1 | G3B2 | G3B3 | G3B4 | G3B5 | G3B6 | G3B7 | G3G1 | G3G2 | G3G3 | G3G4 | G3G5 | G3G6 | G3G7 |
| G4B1 | G4B2 | G4B3 | G4B4 | G4B5 | G4B6 | G4B7 | G4G1 | G4G2 | G4G3 | G4G4 | G4G5 | G4G6 | G4G7 |
| G5B1 | G5B2 | G5B3 | G5B4 | G5B5 | G5B6 | G5B7 | G5G1 | G5G2 | G5G3 | G5G4 | G5G5 | G5G6 | G5G7 |
| G6B1 | G6B2 | G6B3 | G6B4 | G6B5 | G6B6 | G6B7 | G6G1 | G6G2 | G6G3 | G6G4 | G6G5 | G6G6 | G6G7 |
| G7B1 | G7B2 | G7B3 | G7B4 | G7B5 | G7B6 | G7B7 | G7G1 | G7G2 | G7G3 | G7G4 | G7G5 | G7G6 | G7G7 |
P(two_boys | at_least_one_boy) = 1/3"
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "execution_count": 27,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"# Child Problem 2, with days of week enumerated\n",
"Pgrid(S3, 2, two_boys, at_least_one_boy)"
@@ -964,7 +1105,6 @@
"cell_type": "markdown",
"metadata": {
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- "deletable": true,
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"run_control": {
"read_only": false
@@ -978,17 +1118,29 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 28,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "| B1B1 | B1B2 | B1B3 | B1B4 | B1B5 | B1B6 | B1B7 | B1G1 | B1G2 | B1G3 | B1G4 | B1G5 | B1G6 | B1G7 |
| B2B1 | B2B2 | B2B3 | B2B4 | B2B5 | B2B6 | B2B7 | B2G1 | B2G2 | B2G3 | B2G4 | B2G5 | B2G6 | B2G7 |
| B3B1 | B3B2 | B3B3 | B3B4 | B3B5 | B3B6 | B3B7 | B3G1 | B3G2 | B3G3 | B3G4 | B3G5 | B3G6 | B3G7 |
| B4B1 | B4B2 | B4B3 | B4B4 | B4B5 | B4B6 | B4B7 | B4G1 | B4G2 | B4G3 | B4G4 | B4G5 | B4G6 | B4G7 |
| B5B1 | B5B2 | B5B3 | B5B4 | B5B5 | B5B6 | B5B7 | B5G1 | B5G2 | B5G3 | B5G4 | B5G5 | B5G6 | B5G7 |
| B6B1 | B6B2 | B6B3 | B6B4 | B6B5 | B6B6 | B6B7 | B6G1 | B6G2 | B6G3 | B6G4 | B6G5 | B6G6 | B6G7 |
| B7B1 | B7B2 | B7B3 | B7B4 | B7B5 | B7B6 | B7B7 | B7G1 | B7G2 | B7G3 | B7G4 | B7G5 | B7G6 | B7G7 |
| G1B1 | G1B2 | G1B3 | G1B4 | G1B5 | G1B6 | G1B7 | G1G1 | G1G2 | G1G3 | G1G4 | G1G5 | G1G6 | G1G7 |
| G2B1 | G2B2 | G2B3 | G2B4 | G2B5 | G2B6 | G2B7 | G2G1 | G2G2 | G2G3 | G2G4 | G2G5 | G2G6 | G2G7 |
| G3B1 | G3B2 | G3B3 | G3B4 | G3B5 | G3B6 | G3B7 | G3G1 | G3G2 | G3G3 | G3G4 | G3G5 | G3G6 | G3G7 |
| G4B1 | G4B2 | G4B3 | G4B4 | G4B5 | G4B6 | G4B7 | G4G1 | G4G2 | G4G3 | G4G4 | G4G5 | G4G6 | G4G7 |
| G5B1 | G5B2 | G5B3 | G5B4 | G5B5 | G5B6 | G5B7 | G5G1 | G5G2 | G5G3 | G5G4 | G5G5 | G5G6 | G5G7 |
| G6B1 | G6B2 | G6B3 | G6B4 | G6B5 | G6B6 | G6B7 | G6G1 | G6G2 | G6G3 | G6G4 | G6G5 | G6G6 | G6G7 |
| G7B1 | G7B2 | G7B3 | G7B4 | G7B5 | G7B6 | G7B7 | G7G1 | G7G2 | G7G3 | G7G4 | G7G5 | G7G6 | G7G7 |
P(two_boys | at_least_one_boy_tues) = 13/27"
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "execution_count": 28,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"# Child Problem 3\n",
"Pgrid(S3, 2, two_boys, at_least_one_boy_tues)"
@@ -998,7 +1150,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1026,7 +1177,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1049,11 +1199,10 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 29,
"metadata": {
"button": false,
"collapsed": true,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1069,7 +1218,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1086,11 +1234,10 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 30,
"metadata": {
"button": false,
"collapsed": true,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1107,7 +1254,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1119,17 +1265,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 31,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(1, 3)"
+ ]
+ },
+ "execution_count": 31,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"heads = T(\"heads\")\n",
"interviewed = T(\"interviewed\")\n",
@@ -1141,7 +1296,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1157,17 +1311,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 32,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(1, 2)"
+ ]
+ },
+ "execution_count": 32,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(heads, B) "
]
@@ -1176,7 +1339,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1192,7 +1354,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1219,11 +1380,10 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 33,
"metadata": {
"button": false,
"collapsed": true,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1240,7 +1400,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1255,51 +1414,78 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 34,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "{'Car1/Hi/Pick1/Open3', 'Car2/Hi/Pick1/Open3', 'Car2/Lo/Pick1/Open3'}"
+ ]
+ },
+ "execution_count": 34,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"such_that(T(\"Open3\"), M)"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 35,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(1, 3)"
+ ]
+ },
+ "execution_count": 35,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(T(\"Car1\"), such_that(T(\"Open3\"), M))"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 36,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(2, 3)"
+ ]
+ },
+ "execution_count": 36,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(T(\"Car2\"), such_that(T(\"Open3\"), M))"
]
@@ -1308,7 +1494,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1335,11 +1520,10 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 37,
"metadata": {
"button": false,
"collapsed": true,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1356,7 +1540,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1368,45 +1551,72 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 38,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(1, 2)"
+ ]
+ },
+ "execution_count": 38,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(T(\"Car1\"), such_that(T(\"Open3/Goat\"), M2))"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 39,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(1, 2)"
+ ]
+ },
+ "execution_count": 39,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(T(\"Car2\"), such_that(T(\"Open3/Goat\"), M2))"
]
},
{
"cell_type": "code",
- "execution_count": null,
- "metadata": {
- "collapsed": false
- },
- "outputs": [],
+ "execution_count": 40,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Fraction(0, 1)"
+ ]
+ },
+ "execution_count": 40,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"P(T(\"Car3\"), such_that(T(\"Open3/Goat\"), M2))"
]
@@ -1415,7 +1625,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1440,11 +1649,9 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 41,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1475,7 +1682,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1487,17 +1693,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 42,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Counter({False: 33110, True: 66890})"
+ ]
+ },
+ "execution_count": 42,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"from collections import Counter\n",
"\n",
@@ -1506,17 +1721,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 43,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Counter({False: 66738, True: 33262})"
+ ]
+ },
+ "execution_count": 43,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"Counter(monty('stick') for _ in range(10 ** 5))"
]
@@ -1525,7 +1749,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1544,17 +1767,29 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 44,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "{'BB': 0.2645086533229465,\n",
+ " 'BG': 0.24882071317004043,\n",
+ " 'GB': 0.24826679089140383,\n",
+ " 'GG': 0.23840384261560926}"
+ ]
+ },
+ "execution_count": 44,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"DK = ProbDist(GG=121801, GB=126840,\n",
" BG=127123, BB=135138)\n",
@@ -1565,7 +1800,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1577,17 +1811,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 45,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "0.5152805792702689"
+ ]
+ },
+ "execution_count": 45,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"# Child Problem 1 in DK\n",
"P(two_boys, such_that(older_is_a_boy, DK))"
@@ -1595,17 +1838,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 46,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "0.3473082824253857"
+ ]
+ },
+ "execution_count": 46,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"# Child Problem 2 in DK\n",
"P(two_boys, such_that(at_least_one_boy, DK))"
@@ -1615,7 +1867,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1633,11 +1884,9 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 47,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1655,7 +1904,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1667,34 +1915,70 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 48,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "{'BL': 0.00035249828884325804,\n",
+ " 'BN': 0.5146475017111568,\n",
+ " 'GL': 0.0003319644079397673,\n",
+ " 'GN': 0.48466803559206023}"
+ ]
+ },
+ "execution_count": 48,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"child"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 49,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "{'BLBL': 1.2425504363742495e-07,\n",
+ " 'BLBN': 0.00018141236371064045,\n",
+ " 'BLGL': 1.170168857556332e-07,\n",
+ " 'BLGN': 0.0001708446532032245,\n",
+ " 'BNBL': 0.00018141236371064045,\n",
+ " 'BNBN': 0.26486205101753507,\n",
+ " 'BNGL': 0.00017084465320322452,\n",
+ " 'BNGN': 0.24943319367670777,\n",
+ " 'GLBL': 1.170168857556332e-07,\n",
+ " 'GLBN': 0.00017084465320322452,\n",
+ " 'GLGL': 1.102003681388002e-07,\n",
+ " 'GLGN': 0.0001608925374826483,\n",
+ " 'GNBL': 0.0001708446532032245,\n",
+ " 'GNBN': 0.24943319367670777,\n",
+ " 'GNGL': 0.0001608925374826483,\n",
+ " 'GNGN': 0.2349031047246665}"
+ ]
+ },
+ "execution_count": 49,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"S4"
]
@@ -1703,7 +1987,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1715,17 +1998,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 50,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "0.5149145040963757"
+ ]
+ },
+ "execution_count": 50,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"# Child Problem 4\n",
"boy_born_on_leap_day = T(\"BL\")\n",
@@ -1737,7 +2029,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1750,9 +2041,9 @@
"\n",
"> *A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The pot starts at 2 dollars and is doubled every time a head appears. The first time a tail appears, the game ends and the player wins whatever is in the pot. Thus the player wins 2 dollars if a tail appears on the first toss, 4 dollars if a head appears on the first toss and a tail on the second, etc. What is the expected value of this game to the player?*\n",
"\n",
- "To calculate the expected value, we see there is a 1/2 chance of a tail on the first toss (yielding a pot of \\$2) and if not that, a 1/2 × 1/2 = 1/4 chance of a tail on the second toss (yielding a pot of \\$4), and so on. So in total, the expected value is:\n",
+ "To calculate the expected value, we see there is a 1/2 chance of a tail on the first toss (yielding a pot of $2) and if not that, a 1/2 × 1/2 = 1/4 chance of a tail on the second toss (yielding a pot of $4), and so on. So in total, the expected value is:\n",
"\n",
- "$$\\frac{1}{2}\\cdot 2 + \\frac{1}{4}\\cdot 4 + \\frac{1}{8}\\cdot 8 + \\frac{1}{16} \\cdot 16 + \\cdots = 1 + 1 + 1 + 1 + \\cdots = \\infty$$\n",
+ " 2 * (1/2) + 4 * (1/4) + 8 * (1/8) + ... = 1 + 1 + 1 + ... = ∞\n",
"\n",
"The expected value is infinite! But anyone playing the game would not expect to win an infinite amount; thus the paradox.\n",
"\n",
@@ -1763,11 +2054,9 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 51,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1792,7 +2081,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1804,17 +2092,52 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 52,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "{2: 0.5,\n",
+ " 4: 0.25,\n",
+ " 8: 0.125,\n",
+ " 16: 0.0625,\n",
+ " 32: 0.03125,\n",
+ " 64: 0.015625,\n",
+ " 128: 0.0078125,\n",
+ " 256: 0.00390625,\n",
+ " 512: 0.001953125,\n",
+ " 1024: 0.0009765625,\n",
+ " 2048: 0.00048828125,\n",
+ " 4096: 0.000244140625,\n",
+ " 8192: 0.0001220703125,\n",
+ " 16384: 6.103515625e-05,\n",
+ " 32768: 3.0517578125e-05,\n",
+ " 65536: 1.52587890625e-05,\n",
+ " 131072: 7.62939453125e-06,\n",
+ " 262144: 3.814697265625e-06,\n",
+ " 524288: 1.9073486328125e-06,\n",
+ " 1048576: 9.5367431640625e-07,\n",
+ " 2097152: 4.76837158203125e-07,\n",
+ " 4194304: 2.384185791015625e-07,\n",
+ " 8388608: 1.1920928955078125e-07,\n",
+ " 16777216: 5.960464477539063e-08,\n",
+ " 33554432: 2.9802322387695312e-08,\n",
+ " 67108864: 1.4901161193847656e-08,\n",
+ " 100000000: 1.4901161193847656e-08}"
+ ]
+ },
+ "execution_count": 52,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"StP = st_pete(limit=10**8)\n",
"StP"
@@ -1824,7 +2147,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1836,11 +2158,9 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 53,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1856,17 +2176,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 54,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "27.490116119384766"
+ ]
+ },
+ "execution_count": 54,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"EV(StP)"
]
@@ -1875,28 +2204,25 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
"source": [
- "This says that for a casino with a bankroll of 100 million dollars, if you want to maximize your expected value, you should be willing to pay up to \\$27.49 to play the game. Would you pay that much? I wouldn't, and neither would Daniel Bernoulli. \n",
+ "This says that for a casino with a bankroll of 100 million dollars, if you want to maximize your expected value, you should be willing to pay up to $27.49 to play the game. Would you pay that much? I wouldn't, and neither would Daniel Bernoulli. \n",
"\n",
"## Response 2: Value of Money\n",
"\n",
- "Daniel Bernoulli came up with a second response to the paradox based on the idea that if you have a lot of money, then additional money becomes less valuable to you. If I had nothing, and I won \\$1000, I would be very happy. But if I already had a million dollars and I won \\$1000, it would be less valuable. How much less valuable? Bernoulli proposed, and [experiments confirm](https://books.google.com/books?id=1oEa-BiARWUC&pg=PA205&lpg=PA205&dq=mr+beard+oil+wildcatter+value+of+money+utility&source=bl&ots=cBDIX-rkTz&sig=GHB8-inorWrU39vA8JYV_sCtqB8&hl=en&sa=X&ved=0CCAQ6AEwAGoVChMI5fu-p8qlyAIViKWICh0XAAz5#v=onepage&q=mr%20beard%20oil%20wildcatter%20value%20of%20money%20utility&f=false), that *the value of money is roughly logarithmic.* That is, rational bettors don't try to maximize their expected monetary value, they try to maximize their *expected utility*: the amount of \"happiness\" that the money is worth.\n",
+ "Daniel Bernoulli came up with a second response to the paradox based on the idea that if you have a lot of money, then additional money becomes less valuable to you. If I had nothing, and I won $1000, I would be very happy. But if I already had a million dollars and I won $1000, it would be less valuable. How much less valuable? Bernoulli proposed, and [experiments confirm](https://books.google.com/books?id=1oEa-BiARWUC&pg=PA205&lpg=PA205&dq=mr+beard+oil+wildcatter+value+of+money+utility&source=bl&ots=cBDIX-rkTz&sig=GHB8-inorWrU39vA8JYV_sCtqB8&hl=en&sa=X&ved=0CCAQ6AEwAGoVChMI5fu-p8qlyAIViKWICh0XAAz5#v=onepage&q=mr%20beard%20oil%20wildcatter%20value%20of%20money%20utility&f=false), that *the value of money is roughly logarithmic.* That is, rational bettors don't try to maximize their expected monetary value, they try to maximize their *expected utility*: the amount of \"happiness\" that the money is worth.\n",
"I'll write the function `util` to describe what a dollar amount is worth to a hypothetical gambler. `util` says that a dollar is worth a dollar, until the amount is \"enough\" money. After that point, each additional dollar is worth half as much (only brings half as much happiness). Value keeps accumulating at this rate until we reach the next threshold of \"enough,\" when the utility of additional dollars is halfed again. The exact details of `util` are not critical; what matters is that overall money becomes less valuable after we have won a lot of it."
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 55,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1917,7 +2243,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1929,17 +2254,30 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 56,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ " 100 $ = 100 util\n",
+ " 1,000 $ = 1,000 util\n",
+ " 10,000 $ = 4,250 util\n",
+ " 100,000 $ = 15,937 util\n",
+ " 1,000,000 $ = 51,593 util\n",
+ " 10,000,000 $ = 162,460 util\n",
+ " 100,000,000 $ = 535,646 util\n",
+ " 1,000,000,000 $ = 1,658,229 util\n"
+ ]
+ }
+ ],
"source": [
"for d in range(2, 10):\n",
" m = 10 ** d\n",
@@ -1948,17 +2286,33 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 57,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Y axis is util(x); x axis is in thousands of dollars.\n"
+ ]
+ },
+ {
+ "data": {
+ "image/png": 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ZbKdrImIbUCvpwA+5l1mHsX49jBkDX/ximq+yZElKMmadXas69CV9E9gaET8pUHkAWtTy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\n",
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
"source": [
"%matplotlib inline \n",
"import matplotlib.pyplot as plt\n",
@@ -1971,7 +2325,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -1983,11 +2336,9 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 58,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -2003,17 +2354,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 59,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "13.096907431492582"
+ ]
+ },
+ "execution_count": 59,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"EU(StP, util)"
]
@@ -2022,27 +2382,25 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
"source": [
- "That says we should pay up to \\$13.10 to play the game, which sounds more reasonable than \\$27.49.\n",
+ "That says we should pay up to $13.10 to play the game, which sounds more reasonable than $27.49.\n",
"\n",
"# Understanding St. Petersburg through Simulation\n",
"\n",
- "Before I plunk down my \\$13, I'd like to understand the game better. I'll write a simulation of the game:"
+ "Before I plunk down my $13, I'd like to understand the game better. I'll write a simulation of the game:"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 60,
"metadata": {
"button": false,
"collapsed": true,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -2066,7 +2424,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -2078,17 +2435,42 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 61,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "{2: 0.49755,\n",
+ " 4: 0.2506,\n",
+ " 8: 0.1259,\n",
+ " 16: 0.06322,\n",
+ " 32: 0.03151,\n",
+ " 64: 0.01607,\n",
+ " 128: 0.00751,\n",
+ " 256: 0.0037,\n",
+ " 512: 0.00191,\n",
+ " 1024: 0.00106,\n",
+ " 2048: 0.00045,\n",
+ " 4096: 0.00029,\n",
+ " 8192: 0.0001,\n",
+ " 16384: 6e-05,\n",
+ " 32768: 5e-05,\n",
+ " 65536: 1e-05,\n",
+ " 1048576: 1e-05}"
+ ]
+ },
+ "execution_count": 61,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"random.seed(123456)\n",
"\n",
@@ -2100,7 +2482,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -2112,17 +2493,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 62,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(13.2477575, 26.71606)"
+ ]
+ },
+ "execution_count": 62,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"EU(results, util), EV(results)"
]
@@ -2131,7 +2521,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -2145,11 +2534,9 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 63,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -2173,7 +2560,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -2185,17 +2571,26 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 64,
"metadata": {
"button": false,
- "collapsed": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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Ly+uEz6KHe6+yktdKKngyPHtP4NixQ3Xu27xZ3YKD1dcxLGzPx05GhkpZpaSo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\nsJTyTT0IFdJ2YAagA+uBBW92Pg5CuVqAQ2vhFOGcywLgG8AVtfgrgWt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+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
"source": [
"random.seed('running')\n",
"\n",
@@ -2207,26 +2602,25 @@
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- "What can we see from this? Nine of the 10 repetitions have a final expected value payoff (after 100,000 rounds) between 10 and 35. So a price around \\$13 still seems reasonable. One outlier has an average payoff just over 100, so if you are feeling lucky you might be willing to pay more than \\$13.\n",
+ "What can we see from this? Nine of the 10 repetitions have a final expected value payoff (after 100,000 rounds) between 10 and 35. So a price around $13 still seems reasonable. One outlier has an average payoff just over 100, so if you are feeling lucky you might be willing to pay more than $13.\n",
"\n",
"# The Ellsburg Paradox\n",
"\n",
"The [Ellsburg Paradox](https://en.wikipedia.org/wiki/Ellsberg_paradox) has it all: an urn problem; a paradox; a conclusion that can only be resolved through psychology, not mathematics alone; and a colorful history with an inventor, [Daniel Ellsburg](https://en.wikipedia.org/wiki/Daniel_Ellsberg), who went on to become the releaser of the [Pentagon Papers](https://en.wikipedia.org/wiki/Pentagon_Papers). The paradox is as follows:\n",
"\n",
- "> An urn contains 33 red balls and 66 other balls that are either black or yellow. You don't know the mix of black and yellow, just that they total 66. A single ball is drawn at random. You are given a choice between these two gambles:\n",
- "- **R**: Win \\$100 for a red ball.\n",
- "- **B**: Win \\$100 for a black ball.\n",
+ "> An urn contains 33 red balls and 66 other balls that are either black or yellow. You don't know the mix of black and yellow, just that they total 66. A single ball is drawn at random. You are asked which of these two gambles you would prefer:\n",
+ "- **R**: Win 100 for a red ball.\n",
+ "- **B**: Win 100 for a black ball.\n",
"\n",
- "> You are also given a choice between these two gambles:\n",
- "- **RY**: Win \\$100 for a red or yellow ball.\n",
- "- **BY**: Win \\$100 for a black or yellow ball.\n",
+ "> Separately, you are also asked which of these two gambles you prefer:\n",
+ "- **RY**: Win 100 for a red or yellow ball.\n",
+ "- **BY**: Win 100 for a black or yellow ball.\n",
"\n",
"Many people reason as follows: \n",
"- **R**: I win 1/3 of the time\n",
@@ -2235,33 +2629,43 @@
"- **BY**: I win 2/3 of the time. \n",
"- Overall, I prefer the relative certainty of **R** over **B** and of **BY** over **RY**.\n",
"\n",
- "The paradox is that, from an expected utility point of view, that reasoning is inconsistent, no matter what the mix of black and yellow balls is (or no matter what you believe the mix might be). **RY** and **BY** are just the same gambles as **R** and **B**, but with an additional \\$100 for a yellow ball. So if you prefer **R** over **B**, you should prefer **RY** over **BY** (and if you prefer **B** over **R** you should prefer **BY** over **RY**), for any possible mix of black and yellow balls.\n",
+ "The paradox is that, from an expected utility point of view, that reasoning is inconsistent, no matter what the mix of black and yellow balls is (or no matter what you believe the mix might be). **RY** and **BY** are just the same gambles as **R** and **B**, but with an additional 100 for a yellow ball. So if you prefer **R** over **B**, you should prefer **RY** over **BY** (and if you prefer **B** over **R** you should prefer **BY** over **RY**), for any possible mix of black and yellow balls.\n",
"\n",
- "Let's demonstrate. For each possible number of black balls (on the *x* axis), we'll plot the expected value of each of the four gambles; **R** as a solid red line, **B** as a solid black line, **RY** as a dashed red line, and **BY** as a dashed black line:"
+ "Let's demonstrate. For each possible number of black balls (on the *x* axis), we'll plot the expected value of each of the four gambles; **R** as a solid red line, **B** as a solid black line, **RY** as a dotted red line, and **BY** as a dotted black line:"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 65,
"metadata": {
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},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
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3Z+XKldx7770UFhbGHZpIXkslSfzVzMab2f7hxwTgr1EHJgJBN9aRI0cyYsQI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\nSf5XfmvVOO8Q/IJ/EvhGOD0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+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
"source": [
"def ellsburg():\n",
" show('R', 'r')\n",
" show('B', 'k')\n",
- " show('RY', 'r--')\n",
- " show('BY', 'k--')\n",
+ " show('RY', 'r:')\n",
+ " show('BY', 'k:')\n",
" plt.xlabel('Number of black balls')\n",
" plt.ylabel('Expected value of each gamble')\n",
" \n",
- "blacks = list(range(68))\n",
- "urns = [Counter(R=33, B=b, Y=67-b) for b in blacks]\n",
+ "blacks = list(range(67))\n",
+ "all_urns = [Counter(R=33, B=b, Y=66-b) for b in blacks]\n",
" \n",
"def show(colors, line):\n",
- " scores = [score(colors, urn) for urn in urns]\n",
+ " scores = [score(colors, urn) for urn in all_urns]\n",
" plt.plot(blacks, scores, line)\n",
" \n",
"def score(colors, urn): return sum(urn[c] for c in colors)\n",
@@ -2279,25 +2683,34 @@
"\n",
"Similarly, up to 33 black balls, the dashed red line is above the dashed black line, so **RY** is better than **BY**. They are equal at 33, and after that, **BY** is better than **RY**. So in summary, **R** > **B** if and only if **RY** > **BY**.\n",
"\n",
- "It is pretty clear that this hold for every possible mix of black and yellow balls, taken one at a time. But what if you believe that the mix might be one of several possibilities? We'll define `avgscore` to give the score for a gamble (as specified by the colors in it), averaged over a collection of possible urns, each with a different black/yellow mix. Then we'll define `compare` to compare the four gambles on the collection of possible urns:"
+ "It is pretty clear that this holds for every possible mix of black and yellow balls, taken one at a time. But what if you believe that the mix might be one of several possibilities? For example, if we assume that any number of black balls from 0 to 66 is equally likely, then we can use a function, `expected_score` to give the expected return for a gamble (as specified by the colors in the gamble), averaged over a collection of possible urns, each with a different black/yellow mix:"
]
},
{
"cell_type": "code",
- "execution_count": null,
- "metadata": {
- "collapsed": false
- },
- "outputs": [],
+ "execution_count": 66,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "R 33.0\n",
+ "B 33.0\n",
+ "RY 66.0\n",
+ "BY 66.0\n"
+ ]
+ }
+ ],
"source": [
- "def avgscore(colors, urns): \n",
+ "def expected_score(colors, urns): \n",
" return sum(score(colors, urn) for urn in urns) / len(urns)\n",
"\n",
"def compare(urns):\n",
" for colors in ('R', 'B', 'RY', 'BY'):\n",
- " print(colors.ljust(2), avgscore(colors, urns))\n",
+ " print(colors.ljust(2), expected_score(colors, urns))\n",
" \n",
- "compare(urns)"
+ "compare(all_urns)"
]
},
{
@@ -2306,20 +2719,29 @@
"run_control": {}
},
"source": [
- "The above says that if you think any number of black balls is possible and they are all equally equally likely, then you should slightly prefer **B** > **R** and **BY** > **RY**.\n",
+ "This says that **B** and **R** have an equal expected return, as do **BY** and **RY**.\n",
"\n",
- "Now imagine that for some reason you believe that any mix is possible, but that a majority of black balls is more likely (i.e. the urns in the second half of the list of urns are twice as likely as those in the first half). Then we will see that the same preferences hold, but more strongly:"
+ "Now imagine that you believe that any mix is possible, but that a majority of black balls is more likely, in particular that the urns in the second half of the list of `all_urns` are twice as likely as those in the first half. Then we will see that **B** > **R** and **BY** > **RY**:"
]
},
{
"cell_type": "code",
- "execution_count": null,
- "metadata": {
- "collapsed": false
- },
- "outputs": [],
+ "execution_count": 67,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "R 33.0\n",
+ "B 38.554455445544555\n",
+ "RY 60.445544554455445\n",
+ "BY 66.0\n"
+ ]
+ }
+ ],
"source": [
- "compare(urns[:33] + 2 * urns[33:])"
+ "compare(all_urns[:33] + 2 * all_urns[33:])"
]
},
{
@@ -2333,13 +2755,22 @@
},
{
"cell_type": "code",
- "execution_count": null,
- "metadata": {
- "collapsed": false
- },
- "outputs": [],
+ "execution_count": 68,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "R 33.0\n",
+ "B 27.39\n",
+ "RY 71.61\n",
+ "BY 66.0\n"
+ ]
+ }
+ ],
"source": [
- "compare(2 * urns[:33] + urns[33:])"
+ "compare(2 * all_urns[:33] + all_urns[33:])"
]
},
{
@@ -2348,35 +2779,7 @@
"run_control": {}
},
"source": [
- "This time the preferences are reversed for both gambles, **R** > **B** and **RY** > **BY**.\n",
- "\n",
- "Now let's try another approach. Imagine there are two urns, each as described before, and the ball will be drawn from one or the other. We will plot the expected value of each of the four gambles, over all possible pairs of two different urns (sorted by the number of black balls in the pair):"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {
- "collapsed": false,
- "run_control": {}
- },
- "outputs": [],
- "source": [
- "def ellsburg2():\n",
- " show2('R', 'r')\n",
- " show2('B', 'k')\n",
- " show2('RY', 'r--')\n",
- " show2('BY', 'k--')\n",
- " plt.xlabel('Different combinations of two urns')\n",
- " plt.ylabel('Expected value of each gamble')\n",
- " \n",
- "def show2(colors, line):\n",
- " urnpairs = [(u1, u2) for u1 in urns for u2 in urns]\n",
- " urnpairs.sort(key=lambda urns: avgscore('B', urns))\n",
- " X = list(range(len(urnpairs)))\n",
- " plt.plot(X, [avgscore(colors, urns) for urns in urnpairs], line)\n",
- " \n",
- "ellsburg2()"
+ "This time the preferences are reversed for both gambles, **R** > **B** and **RY** > **BY**."
]
},
{
@@ -2385,9 +2788,7 @@
"run_control": {}
},
"source": [
- "The curves are different, but the results are the same: **R** > **B** if and only if **RY** > **BY**.\n",
- "\n",
- "So why do many people prefer **R** > **B** and **BY** > **RY**. One explanation is *risk aversion*; it feels safer to take a definite 1/3 chance of winning, rather than a gamble that might be as good as 2/3, but might be as bad as 0. This is irrational thinking (in the sense that those who follow this strategy will win less), but people are sometimes irrational."
+ "So why do many people prefer **R** > **B** and **BY** > **RY**? One explanation is *risk aversion*; it feels safer to take a definite 1/3 chance of winning, rather than a gamble that might be as good as 2/3, but might be as bad as 0. This is irrational thinking (in the sense that those who follow this strategy will win less), but people are sometimes irrational."
]
},
{
@@ -2405,13 +2806,11 @@
},
{
"cell_type": "code",
- "execution_count": null,
- "metadata": {
- "collapsed": false
- },
+ "execution_count": 69,
+ "metadata": {},
"outputs": [],
"source": [
- "# Good and bad outcomes for kidney stone reatments A and B,\n",
+ "# Good and bad outcomes for kidney stone treatments A and B,\n",
"# each in two cases: [small_stones, large_stones]\n",
"A = [Counter(good=81, bad=6), Counter(good=192, bad=71)]\n",
"B = [Counter(good=234, bad=36), Counter(good=55, bad=25)]\n",
@@ -2430,23 +2829,42 @@
},
{
"cell_type": "code",
- "execution_count": null,
- "metadata": {
- "collapsed": false
- },
- "outputs": [],
+ "execution_count": 70,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[0.9310344827586207, 0.7300380228136882]"
+ ]
+ },
+ "execution_count": 70,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"[success(case) for case in A]"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 71,
"metadata": {
- "collapsed": false,
"run_control": {}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[0.8666666666666667, 0.6875]"
+ ]
+ },
+ "execution_count": 71,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"[success(case) for case in B]"
]
@@ -2466,24 +2884,44 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 72,
"metadata": {
- "collapsed": false,
"run_control": {}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "0.78"
+ ]
+ },
+ "execution_count": 72,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"success(A[0] + A[1])"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 73,
"metadata": {
- "collapsed": false,
"run_control": {}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "0.8257142857142857"
+ ]
+ },
+ "execution_count": 73,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"success(B[0] + B[1])"
]
@@ -2494,16 +2932,15 @@
"run_control": {}
},
"source": [
- "Overall, **B** is more successful, 83% to 78%, even though **A** is better in both cases. So if you had kidney stones, and you want the highest chance of success, which treatment would you prefer? If you knew you had small stones (or large stones), the evidence supports **A**. But if the size was unknown, does that mean you should prefer **B**? Analysts agree that the answer is no, you should stick with **A**. The only reason why **B** has a higher overall success rate is that doctors choose to do **B** more often on the easier, small stone cases, and reserve **A** for the harder, large stone cases. **B** is better, but it has a lower overall percentage because it is given the difficult patients.\n",
+ "Overall, **B** is more successful, 83% to 78%, even though **A** is better in both cases. So if you had kidney stones, and you want the highest chance of success, which treatment would you prefer? If you knew you had small stones (or large stones), the evidence supports **A**. But if the size was unknown, does that mean you should prefer **B**? Analysts agree that the answer is no, you should stick with **A**. The only reason why **B** has a higher overall success rate is that doctors choose to do **B** more often on the easier, small stone cases, and reserve **A** for the harder, large stone cases. **A** is better, but it has a lower overall percentage because it is given the difficult patients.\n",
"\n",
- "Here's another example, showing the batting averages for two baseball players, Derek jeter and David Justice, for the years 1995 and 1996:"
+ "Here's another example, showing the batting averages for two baseball players, Derek Jeter and David Justice, for the years 1995 and 1996 (I should say that Justice is considered a very good player, but Jeter is considered even better, a sure-bet future Hall of Fame player):"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 74,
"metadata": {
- "collapsed": true,
"run_control": {}
},
"outputs": [],
@@ -2511,29 +2948,49 @@
"Jeter = [Counter(hit=12, out=36), Counter(hit=183, out=399)]\n",
"Justice = [Counter(hit=104, out=307), Counter(hit=45, out=95)]\n",
"\n",
- "def BA(case): \"Batting average\"; return ProbDist(case)['hit'] "
+ "def BA(case): \"Batting average\"; return ProbDist(case)['hit']"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 75,
"metadata": {
- "collapsed": false,
"run_control": {}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[0.25, 0.31443298969072164]"
+ ]
+ },
+ "execution_count": 75,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"[BA(year) for year in Jeter]"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 76,
"metadata": {
- "collapsed": false,
"run_control": {}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "[0.25304136253041365, 0.32142857142857145]"
+ ]
+ },
+ "execution_count": 76,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"[BA(year) for year in Justice]"
]
@@ -2549,24 +3006,44 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 77,
"metadata": {
- "collapsed": false,
"run_control": {}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "0.30952380952380953"
+ ]
+ },
+ "execution_count": 77,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"BA(Jeter[0] + Jeter[1])"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 78,
"metadata": {
- "collapsed": false,
"run_control": {}
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "0.27041742286751363"
+ ]
+ },
+ "execution_count": 78,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"BA(Justice[0] + Justice[1])"
]
@@ -2586,7 +3063,6 @@
"cell_type": "markdown",
"metadata": {
"button": false,
- "deletable": true,
"new_sheet": false,
"run_control": {
"read_only": false
@@ -2618,9 +3094,9 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
- "version": "3.5.1"
+ "version": "3.5.3"
}
},
"nbformat": 4,
- "nbformat_minor": 0
+ "nbformat_minor": 1
}