How many people need to be in the same room for there to be a 50% chance that two of those people will have the same birthday?
Answer
23 people
Analysis
When working out this kind of problem, it's all in the approach. For this problem, let's work up to a method of doing it.
One of to do it would be to find the number directly - as in calculating the probability "from the ground up". The other way to do it is to see what the probability is of not having the situation and subtracting that from 1 (1 being the sum of all probabilities possible). It's this "subtracting" approach we'll use. To see how, let's follow smaller groups of people and work up to bigger ones.
Let's say we have 1 person in a room (person A). A can have a birthday that can be any one of 365 days (we'll ignore leap days). We can express this as
If we put a second person into the room (person B), B too can have any one of 365 days as their birthday. If it's not the same day as A, then it can be any one of 364 days:
probability that it's different.
Which also means that there is a
probability it's the same.
If we put a third person into the room (person C), C can also have any of the 365 days as their birthday. For A, B, and C to all have different birthdays, we can say:
probability that they are all different, which means there's a 0.0083 probability (1 - 0.9917) one of them is the same.
And so on. Note that we end up with a numerator that is a permutation and a denominator that is 365 taken to the power of the number of people in the room. Therefore, we can generalize the calculation as:
And now we want to find where our expression is 50%:
This is a very gnarly problem to work with algebra - it'd be better to use a graphing technique or a spreadsheet to work through values of n. And so I worked it using a spreadsheet and found that at 23 people, there's roughly a 50% probability of at least 1 shared birthday.
As a side note, at 57 people, there is a better than 99% probability of at least 1 shared birthday.
The spreadsheet is here:
https://docs.google.com/spreadsheets/d/1XPWcgzaXS5zcXOF5M0VzofFMxJGtsFwlrrX_3ybPxtE/edit?usp=sharing
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