(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 137146, 3008] NotebookOptionsPosition[ 113445, 2519] NotebookOutlinePosition[ 131705, 2836] CellTagsIndexPosition[ 131662, 2833] WindowFrame->Normal ContainsDynamic->True *) (* Beginning of Notebook Content *) Notebook[{ Cell["The Birthday Bet", "Title"], Cell[TextData[ButtonBox["Mark Senn", BaseStyle->"Hyperlink", ButtonData->{ URL["http://engineering.purdue.edu/~mark"], None}]], "Text", CellChangeTimes->{{3.395091778499296*^9, 3.395091781665388*^9}, { 3.395091905730947*^9, 3.395091905734182*^9}}, FontSize->24], Cell[TextData[StyleBox["created December 13, 2006\nrevised September 27, \ 2008", FontSize->24, FontVariations->{"CompatibilityType"->0}, FontColor->GrayLevel[0]]], "Subsubtitle", CellChangeTimes->{{3.395090977172123*^9, 3.395091006169457*^9}, { 3.431477115185335*^9, 3.431477135297826*^9}}, FontSize->24], Cell["\<\ In a class of 23 students there is a 50.8% probability that two or more \ students have the same birthday. This is a result from the classic \ statistics \"birthday problem\" (Dr. Math, 2006; Reese, 2005; Weisstein \ 2003). (Year of birth, leap years, and non-uniform distribution of birthday \ issues are ignored.)\ \>", "Text", CellMargins->{{19.25, Inherited}, {Inherited, Inherited}}, CellChangeTimes->{{3.395090591510066*^9, 3.395090594478733*^9}}], Cell[CellGroupData[{ Cell["Birthday Combinations", "Section"], Cell[TextData[{ "First, for warm-up, how many possible combinations of birthdays are there \ for people in the class? For one person there are 365 possibilities For two \ people, ", StyleBox["365 \[Times] ", "InlineFormula"], "365 combinations. For ten people" }], "Text", CellMargins->{{19.25, Inherited}, {Inherited, Inherited}}], Cell[CellGroupData[{ Cell["365 ^ 10", "Input", CellLabel->"In[6]:="], Cell[BoxData["41969002243198805166015625"], "Output", CellChangeTimes->{3.431482734032061*^9}, CellLabel->"Out[6]="] }, Open ]], Cell[TextData[{ "combinations. For ", Cell[BoxData[ FormBox["n", TraditionalForm]]], StyleBox[" ", FontSlant->"Italic"], StyleBox["people and ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox["d", TraditionalForm]]], StyleBox[" ", "DisplayFormula", FontVariations->{"CompatibilityType"->0}], StyleBox["days, there are ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SuperscriptBox["d", "n"], TraditionalForm]], "InlineFormula"], StyleBox[" possibilites.", "InlineFormula", FontVariations->{"CompatibilityType"->0}] }], "Text", CellChangeTimes->{3.431478955822517*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"d", "^", "n"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"d", ",", " ", "365", ",", " ", "\"\\""}], "}"}], ",", " ", "1", ",", "1000", ",", " ", "1", ",", " ", RowBox[{"Appearance", "\[Rule]", "\"\\""}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"n", ",", "10", ",", " ", "\"\\""}], "}"}], ",", "1", ",", "1000", ",", " ", "1", ",", " ", RowBox[{"Appearance", "\[Rule]", "\"\\""}]}], "}"}]}], "\[IndentingNewLine]", "]"}]], "Input", CellChangeTimes->{{3.431479012271825*^9, 3.431479019414283*^9}, { 3.431479752092823*^9, 3.431479791869282*^9}, {3.4314798429803333`*^9, 3.431479859587721*^9}, 3.431480911023572*^9, {3.43148097265807*^9, 3.4314810144117002`*^9}}, CellLabel->"In[7]:="], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`d$$ = 365, $CellContext`n$$ = 10, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{{ Hold[$CellContext`d$$], 365, "d days"}, 1, 1000, 1}, {{ Hold[$CellContext`n$$], 10, "n people"}, 1, 1000, 1}}, Typeset`size$$ = {459.875, {1., 18.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`d$631$$ = 0, $CellContext`n$632$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`d$$ = 365, $CellContext`n$$ = 10}, "ControllerVariables" :> { Hold[$CellContext`d$$, $CellContext`d$631$$, 0], Hold[$CellContext`n$$, $CellContext`n$632$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> $CellContext`d$$^$CellContext`n$$, "Specifications" :> {{{$CellContext`d$$, 365, "d days"}, 1, 1000, 1, Appearance -> "Open"}, {{$CellContext`n$$, 10, "n people"}, 1, 1000, 1, Appearance -> "Open"}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{511., {125.5625, 137.4375}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{3.431482735042762*^9}, CellLabel->"Out[7]="] }, Open ]], Cell[" So, for 365 days and 23 people there are", "Text", CellChangeTimes->{3.431479001240592*^9}], Cell[CellGroupData[{ Cell["\<\ d = 365; n = 23; d^n\ \>", "Input", CellLabel->"In[8]:="], Cell[BoxData["85651679353150321236814267844395152689354622364044189453125"], \ "Output", CellChangeTimes->{3.431482735435306*^9}, CellLabel->"Out[10]="] }, Open ]], Cell[TextData[StyleBox["combinations.", "Text"]], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Birthdays on Same Day", "Section"], Cell[TextData[{ "How likely is it that two or more people in a group of ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " people have their birthday on the same month and day?" }], "Text"], Cell["The Hard Way", "Subsection"], Cell[TextData[{ "One can find the probability of two people, three people, and so on up to \ all ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " people having a birthday (Clough, 2006). This is complicated, see ", StyleBox["Weisstein (2003", FontFamily->"Times New Roman"], StyleBox[") ", FontFamily->"Lucida Grande"], " for details." }], "Text"], Cell["The Easy Way", "Subsection"], Cell[TextData[{ "An easier way to solve the problem is to find the probability that everyone \ in the class has separate birthdays and subtract that from 1. The first \ person has ", Cell[BoxData[ FormBox[ RowBox[{"d", "/", "d"}], TraditionalForm]]], " ways to have a birthday, the second ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"d", "-", "1"}], ")"}], "/", " ", "d"}], TraditionalForm]]], " ways to have a birthday on a different day than the first, the third has \ ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"d", "-", "2"}], ")"}], "/", "d"}], TraditionalForm]]], " ways to have a birthday on a different day than the first and second, etc. \ So, doing the math manually, the probability of everyone having a different \ birthday is" }], "Text"], Cell[BoxData[{ FormBox[ RowBox[{ FractionBox[ RowBox[{ RowBox[{"d", RowBox[{"(", RowBox[{"d", "-", "1"}], ")"}]}], RowBox[{"\[CenterEllipsis]", RowBox[{"(", RowBox[{"d", "-", "n", "+", "1"}], ")"}]}]}], SuperscriptBox["d", "n"]], "\n"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{"=", " ", FormBox[ RowBox[{ FractionBox[ RowBox[{ RowBox[{"d", "(", RowBox[{"d", "-", "1"}], ")"}], RowBox[{"\[CenterEllipsis]", "(", RowBox[{"d", "-", "n", "+", "1"}], ")"}], RowBox[{ RowBox[{"(", RowBox[{"d", "-", "n"}], ")"}], "!"}]}], RowBox[{ RowBox[{ SuperscriptBox["d", "n"], "(", RowBox[{"d", "-", "n"}], ")"}], "!"}]], " "}], TraditionalForm]}], "\n"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{"=", RowBox[{ FractionBox[ RowBox[{"d", "!"}], RowBox[{ RowBox[{ SuperscriptBox["d", "n"], "(", RowBox[{"d", "-", "n"}], ")"}], "!"}]], ".", "\[IndentingNewLine]"}]}], TraditionalForm]}], "DisplayFormula", CellChangeTimes->{{3.431477276329681*^9, 3.4314774961351357`*^9}, 3.4314828578220797`*^9}], Cell[TextData[{ "Or, ", StyleBox["Mathematica", FontSlant->"Italic"], " can be used to do the math (Agrawal, 2006)." }], "Text"], Cell[CellGroupData[{ Cell["\<\ Clear[d,n]; expr = Product[d-i, {i,0,n-1}] / d^n\ \>", "Input", CellChangeTimes->{{3.3951297867325087`*^9, 3.395129797978507*^9}, { 3.3951299370169153`*^9, 3.395129940247007*^9}, {3.395130083779016*^9, 3.395130091129236*^9}, {3.395130132011384*^9, 3.395130132600606*^9}}, CellLabel->"In[11]:="], Cell[BoxData[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], "n"], " ", SuperscriptBox["d", RowBox[{"-", "n"}]], " ", RowBox[{"Pochhammer", "[", RowBox[{ RowBox[{"-", "d"}], ",", "n"}], "]"}]}]], "Output", CellChangeTimes->{3.431482736486712*^9}, CellLabel->"Out[12]="] }, Open ]], Cell[CellGroupData[{ Cell["\<\ fexp = FunctionExpand[ expr, d > 0 && n > 0 && Element[{d,n}, Integers] ]\ \>", "Input", CellLabel->"In[13]:="], Cell[BoxData[ FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"-", "1"}], ")"}], RowBox[{"2", " ", "n"}]], " ", SuperscriptBox["d", RowBox[{"-", "n"}]], " ", RowBox[{"Gamma", "[", RowBox[{"1", "+", "d"}], "]"}]}], RowBox[{"Gamma", "[", RowBox[{"1", "+", "d", "-", "n"}], "]"}]]], "Output", CellChangeTimes->{3.431482737621307*^9}, CellLabel->"Out[13]="] }, Open ]], Cell[TextData[{ "Mathematica apparently doesn't know that Gamma[x] can be simplified to \ (x-1)! 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The probability of the second person having the \ same birthday as first should be 1/365.\ \>", "Text"], Cell[CellGroupData[{ Cell["p3", "Input", CellLabel->"In[20]:="], Cell[BoxData[ FractionBox["1093", "133225"]], "Output", CellChangeTimes->{3.431482740451066*^9}, CellLabel->"Out[20]="] }, Open ]], Cell[CellGroupData[{ Cell["N[p3]", "Input", CellLabel->"In[21]:="], Cell[BoxData["0.008204165884781385`"], "Output", CellChangeTimes->{3.431482740941987*^9}, CellLabel->"Out[21]="] }, Open ]], Cell["Looks reasonable for three people.", "Text"], Cell[CellGroupData[{ Cell["p365", "Input", CellLabel->"In[22]:="], Cell[BoxData[ RowBox[{"51517032665812765302327026893496883186400665697005980370247176268541\ 373837192267654095571925517603383762830631582518187715074175923873148793677521\ 219003356406056521703165087092672607628558513424411433898362967347460375892253\ 010565149603837309299743976790111410992046109088069097592817509861890413283779\ 336530891010889660626950329304171445982818893538415634607303335759477166903038\ 881662594835361773495775281138585652712431903790211718125087251308803244967805\ 015947293528973905245466698337129457874723878837489105746799000399224588719517\ 216077620373978913694431126327074837950731731357336279886839001643041582018672\ 598935497031627610704056498361936708786676369701291556370213413941137182966060\ 728097866053527855440507456941424880897170425077573153185020367461413306977657\ 783631970457070371091697159174482599456528796315906309510553614217672220870601\ 8846440325979761", "/", "515170326658127653023270268934968831864006656970059803702471762685413738371\ 922676540955719255176033837628306315825181877150741759238731487936775212190033\ 564135520192401974269458851863217125548343091322801534117245573443413531922275\ 612913528038368117246368938917573264013996463001304954829864348817232466436717\ 311856873167200871359539951780183986982019981454546048785471012647019662520753\ 069879400574056837803950541557117787442934714835950267467275488609709127945889\ 546366868819498143130928828104709275844760637935840192366628163101930811267397\ 402954354789999800088756824182179122318041220995650232069431164178955161360483\ 122690952293589731983424333163050451909518452256437800568549944499327426042068\ 267202215812972708997217666469620959661388470725356466960622014888773469229307\ 515572285821265768570192890040020733139958437375727218210386126884259283542633\ 056640625"}]], "Output", CellChangeTimes->{3.4314827417049026`*^9}, CellLabel->"Out[22]="] }, Open ]], Cell["Show numerical value to 100 digit precision.", "Text"], Cell[CellGroupData[{ Cell["N[p365,100]", "Input", CellLabel->"In[23]:="], Cell[BoxData["1.\ 000000000000000000000000000000000000000000000000000000000000000000000000000000\ 0000000000000000000000000000000000054`100."], "Output", CellChangeTimes->{3.4314827420275373`*^9}, CellLabel->"Out[23]="] }, Open ]], Cell["\<\ The probability of two people having the same birthday in a group of 365 \ people being one doesn't make any sense. Everyone could have a different \ birthday and it would require 366 people to have two people with the same \ birthday. Maybe printing with 500 digit precision will give a clue what's \ going on.\ \>", "Text"], Cell[CellGroupData[{ Cell["N[p365,500]", "Input", CellLabel->"In[24]:="], Cell[BoxData["0.\ 999999999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999999999\ 854504478438129659662859840961474839950577983539826519536685473165935984059430\ 273456160893081452012351538951270248496967913253518501428929192857660287347571\ 775847288152296739381622109901909624814813131416419809001306584764345305734242\ 722465159164380409404828588904635390273536762068031488876787398142736913193823\ 823800330733937620637194828670356943176055`500."], "Output", CellChangeTimes->{3.431482742690633*^9}, CellLabel->"Out[24]="] }, Open ]], Cell["Aha! It was almost one, but not quite.", "Text"], Cell[CellGroupData[{ Cell["p366", "Input", CellLabel->"In[25]:="], Cell[BoxData["1"], "Output", CellChangeTimes->{3.431482743060857*^9}, CellLabel->"Out[25]="] }, Open ]], Cell["\<\ This makes sense, with 366 people at least two people are guaranteed to have \ the same birthday.\ \>", "Text"], Cell["\<\ What is the probability that exactly 366 people will be required? \ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ p366365 = f[d,366] - f[d,365]; N[p366365]\ \>", "Input", CellLabel->"In[26]:="], Cell[BoxData["1.4549552156187035`*^-157"], "Output", CellChangeTimes->{3.4314827433097343`*^9}, CellLabel->"Out[27]="] }, Open ]], Cell["So, 366 people are needed", "Text"], Cell[CellGroupData[{ Cell["N[1/p366365]", "Input", CellLabel->"In[28]:="], Cell[BoxData["6.873063784130022`*^156"], "Output", CellChangeTimes->{3.4314827440714607`*^9}, CellLabel->"Out[28]="] }, Open ]], Cell[TextData[{ "about 7 out of every ", Cell[BoxData[ FormBox[ SuperscriptBox["10", "156"], TraditionalForm]]], " times. Aside: there are around ", Cell[BoxData[ FormBox[ SuperscriptBox["10", "80"], TraditionalForm]]], " atoms in the universe [Champion, 1998; Wikipedia, 2006] so the probability \ of choosing a specific atom from the universe at random (talk about a needle \ in a haystack!) is much better than choosing a group of 365 random people \ (remember this is a mathematical model and there is an inexhaustible supply \ of people) having different birthdays." }], "Text"], Cell["Make Some Plots", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"f", "[", RowBox[{"d", ",", "n"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"n", ",", "2", ",", RowBox[{"d", "+", "1"}]}], "}"}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"d", ",", " ", "365", ",", " ", "\"\\""}], "}"}], ",", " ", "1", ",", " ", "365", ",", " ", "1", ",", " ", RowBox[{"Appearance", "->", "\"\\""}]}], "}"}]}], 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