The Birthday Problem - explanation

Let's look at the probabilities a step at a time.

For one person, there are 365 distinct birthdays.
For two people, there are 364 different ways that the second could have a birthday without matching the first.
If there is no match after two people, the third person has 363 different birthdays that do not match the other two. So, the probability of a match is 1 - (365)(364)(363)/(365)(365)(365).
This leads to the following formula for calculating the probability of a match with N birthdays is 1 - (365)(364)(363)...(365 - N + 1)/(365)^N.

Running this through a computer gives the chart below. Notice that a probability of over .5 is obtained after 23 dates!

Chart of birthday probabilities with Number of Birthdays along X-axis and Probability of a Match along Y-axis. The graph curves up, rising quickly from 0,0 to 50% probability by 23 people, and slowly leveling off after about 40 people at above 90% probability. By 60 people, the probability is nearly 100%.
Notice that the probability is above .9 before the sample size reaches even 45.

Return to the Introduction.