Observations and results
Did about 50 percent of the groups of 23 or more people include at least two people with the same birthdays?
When comparing probabilities with birthdays, it can be easier to look at the probability that people do not share a birthday. A person's birthday is one out of 365 possibilities (excluding February 29 birthdays). The probability that a person does not have the same birthday as another person is 364 divided by 365 because there are 364 days that are not a person's birthday. This means that any two people have a 364/365, or 99.726027 percent, chance of not matching birthdays.
As mentioned before, in a group of 23 people, there are 253 comparisons, or combinations, that can be made. So, we're not looking at just one comparison, but at 253 comparisons. Every one of the 253 combinations has the same odds, 99.726027 percent, of not being a match. If you multiply 99.726027 percent by 99.726027 253 times, or calculate (364/365)253, you'll find there's a 49.952 percent chance that all 253 comparisons contain no matches. Consequently, the odds that there is a birthday match in those 253 comparisons is 1 – 49.952 percent = 50.048 percent, or just over half! The more trials you run, the closer the actual probability should approach 50 percent.
More to explore
"Understanding the Birthday Paradox" from BetterExplained
"Probability Central" from Oracle ThinkQuest
"Combinations and Permutations" from MathIsFun
"The Birthday Paradox" from Science Buddies
This activity brought to you in partnership with Science Buddies