# Birthday problem with 3 people

I have the following problem. It is a simple birthday probability problem with 3 people but I can't crack it

Annie, Boris, and Charlie have random and independent birthdays. (We ignore leap years, so that a year has 365 days.) What is the probability that Annie, Boris, and Charlie have the same birthday?

My attempt is to take the probability that 2 of them have the same birthday and multiply it by 3, because there are three people but this does not seem to be right.

• Suppose Annie's birthday is April 15. What is the probability that two randomly chosen people have April 15 as their birthdays? – zhw. Apr 15 '15 at 18:13
• Asking that three people have the same birthday is (much) more restrictive than asking that two people have the same birthday, so the probability must be less. Multiplying by three is very close if you want the chance that some pair of them share a birthday as there are three pairs. It fails because you triple count the cases where they all have the same birthday. – Ross Millikan Apr 15 '15 at 18:16

To solve this problem, take it step by step.

What is the probability of Annie having a birthday on a particular day of the year: 1/365

What is the probability of Boris having a birthday on the same day = (1/365) * (1/365)

Similarly, probability of Charlie having a birthday on the same day = (1/365)^3

The above answer is for a specific day in a year. Since we are fine with any day in the year, multiply the answer with 365 (total number of days in the year).

So, probability of all three having a birthday on the same day in the year = (1/365)^2

take a particular day , each of them have a probability of having their birthday on that day as $\frac{1}{365}$ now thus all of them having their probability on that particular day is ${(\frac{1}{365})}^3$ now there are $365$ days thus total probability is $365\cdot{(\frac{1}{365})}^3={(\frac{1}{365})}^2$

Since the probability that Annie will have the same birthday as herself is $1$, another way of phrasing the question is to ask "What is the probability that both Boris and Charlie have the same birthday as Annie?" The probability that Boris will share her birthday is $1/365$. Likewise, the probability that Charlie will share Annie's birthday is $1/365$. Since the dates of their birthdays are independent, the probability that both Boris and Charlie will have the same birthday as Annie is $$1 \cdot \frac{1}{365} \cdot \frac{1}{365} = \left(\frac{1}{365}\right)^2$$