12th grade level
if you are at a party what is the least amount of people that have to be there to have at least a 50% chance of having 2 people with the same birthday? and how do you get the answer thank you in advance
I want to first find p, the probability that in a room with k people, NO two have the same birthday. The probability that two people have the same birthday is then 1-p. Ignoring leap years, there are 365k possibilities for the birthdays of k people. If these are people selected at random then each of the possibilities is equally likely.
We now want to count the number of possibilities for birthdays with no day appearing twice. The first person can have any of the 365 days as his/her birthday. Now the second person is restricted to 364 days if he/she is to have a birthday on a different day. Thus the number of possibilities for the first two people is 365x364. The third person has only 363 days available so the number of possibilities for the first three people is 365x364x363. Continuing on we see that the probability that no two of the k people have the same birthday is
To answer your question you need to do some arithmetic to find the value of k that makes 1-p approximately 1/2.
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