Birthday problem
Probability
Level
3
There are 23 people in a room. What is the probability of at least 2 people sharing a birthday? (Ignoring February 29 and considary all 365 days to be equally probable)
69.3
50.7
6.3
12.3
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"at least 2 people having the same birthday" includes 3 people sharing one all the way up to everyone having the same birthday so therefore the complement is that no one has the same birthday. So we can calculate 1-p(no one shares a birthday). The potential birthdays person 1 can have is 365/365. Person 2 can have 364/365 since person 2 cannot share the missing day with person 1. Likewise person 3 can have days 363/365 as to not share days 364 and 365 with persons 1 and 2. Continuing this process we get person 23 with potential days 343/365. From this we can multiply these independent events to get 1-(364!/342!)/(365^22)=.507. This is simplified to not include person 1, whose probability is 1.