The Birthday Problem in a Million Simulations

Simulate and analyze the probability of shared birthdays in a million randomly selected groups

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According to the Birthday Problem, if birthdays are uniformly distributed across a 365-day year, a group of just 23 people has approximately a 50.7% chance of at least one shared birthday.

Generate a list of all possible days in a non-leap year:

In[1]:=

aYear=DayRange[{2007,1,1},{2007,12,31}];

Format the output to display only the month and day, removing the year:

In[2]:=

birthdays=DateString[#,{"MonthName"," ","DayShort"}]&/@aYear;birthdays//Short

Out[2]//Short=

{January 1,January 2,January 3,359,December 29,December 30,December 31}

Simulate one million random groups of 23 people from the formatted birthday list:

In[3]:=

millionGroups=Table[RandomChoice[birthdays,23],1000000];

Detect the existence of shared birthdays within each group, returning 1 if there is a match and 0 if there is not:

In[4]:=

results=Boole[Not[DuplicateFreeQ[#]]]&/@millionGroups;

Count occurrences of groups with and without shared birthdays:

In[5]:=

counts=Counts[results]

Out[5]=

1507073,0492927

Calculate the probability of shared birthdays across groups:

In[6]:=

probability=N[counts[1]/Total[counts]]

Out[6]=

0.507073

Create a histogram to visualize the result:

In[7]:=

Histogramresults,



Out[7]=

The simulation confirms the result of the Birthday Problem: in a million random groups of 23 people, approximately 50.7% of the groups had at least one shared birthday, closely matching the theoretical probability.

External Links

Birthday Problem Calculator
The Birthday Problem and Some Generalizations
The Birthday Problem

Source Metadata

Citation:
- Birthday Problem: Wolfram MathWorld
- Birthday Problem: Better Explained
- Birthday Problem: Wikipedia

Related Symbols

DayRange
DateString
RandomChoice
DuplicateFreeQ
Boole

Publisher Information

Contributed by: Akram Masoud