# Probability that at Least 2 People Share the Same Birthday

by on September 27th, 2010

To find the probability that at least 2 people share the same birthday computer simulation is used. Creating random numbers between 1 and 365 for 25, 20, 75, 100, 500, and 1000 sample sizes in Excel then using the output, a frequency table is created for each trial. The simulation is then compared to the probability that at least 2 people share the same birthday if Complementary Probability is used.

For the simulation we are using the following assumptions:

Every day is equally likely for a birthday to occur. Each birthday is considered independent of any other birthday occurring, not mutually exclusive. Leap years are not taken into account so we are using 365 days a year. Multiple births would share the same birthday. Birth years are not considered, only the month and day. Outside factors are not taken into account which would change the frequency of births. Events like power outages and snow storms which would influence the number of births nine months later. The simulation uses Excel’s RANDBETWEEN(1,365) to randomly create each birthday between 1 and 365. To count the number of birthdays greater than 1 for any particular day for each trial the COUNTIFS(C1:C25,”>1″) function is used. The Experimental Probability shows that a sample size of as few as 25 people there is an 80% probability that at least 2 people share the same birthday. When increasing the sample size to 50 the probability increases to 100%.

See Probability Table 1

Using Complementary Probability the Probability that at least 2 people share the same birthday in a sample size of 25 people is 56.9% compared to the simulation with 80% probability. With each increase in the sample size the Experimental probability is close or equal to the Complementary Probability.

See Probability Table 2

Therefore, the probability that at least 2 people share the same birthday increases as the number of people increase. For a group of 25 people there is a 56.9-80% chance that at least 2 or more people share the same birthday. When a group of 75 people are used the probability of at least 2 or more people share the same birthday is 100% for both the experimental and the complementary probability.

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