Birthday Probability

(Sheldon Ross, 1988, A First Course in Probability, 3^rd edition, Macmillan Publishing Company, New York, NY, pgs. 160-162.)

 The probability of an event (E) is the number of ways the event can occur over all of the possible outcomes.  All possible outcomes that can occur are known as the sample space.  The number of ways the event can occur is a subset of the sample space (S).  The equation for the probability of an event is: 

P(E) = number of outcomes in E .

number of outcomes in S

The easiest way to calculate the probability of two people in the room having the same birthday is to first calculate the probability that no two people in the room have the same birthday and subtract this from one (based on one of the propositions of probability that states that all possible probabilities in an event must sum to one).  To calculate the probability that no two people in the room will have the same birthday, we first need to know the number of people in the room (n).  Since each person can celebrate his/her birthday on any of 365 days, the total number of possible outcomes for those n people is (365)ⁿ (this is the sample space and the denominator of the equation). The numerator is the number of outcomes for the event and is equal to (365)(364)(363)…(365-n+1).  Since no two people in the event can have a birthday on the same day, the first person can have his/her birthday on any of the 365 days in a year.  The second person can only have their birthday on any of the remaining 364 days in a year.  The third person can have their birthday on any of the remaining 363 days in a year, and so on.  The total number of possibilities in the event is the product of all of these.  Therefore, the probability that at least two people in the room celebrate the same birthday is:

P( > 2 same birthdays) =

For example, when there are 23 people in the room, there is just over a 50% chance that two or more people in the room celebrate their birthday on the same day.  In fact, the following table can be used to know the probability of this occurring if the number of persons in the room is known;

 Probability of 2 or more birthdays on the same day

Number of people in the room

 50.73%

23

 87.82%

39

 89.12%

40

 90.32%

41 

91.40%

42

92.39%

43

 93.29%

44

 94.10%

45

 94.82%

46

 94.57%

47

 96.09%

48

 96.57%

49

 97.04%

50