# Birthday probability problem

There are four people in a room, namely P, Q, R and S.

Q's birthday is different from everyone else. What is the probability that P and R share the same birthday?

I'm getting $1/364$ as answer. $(365*364*1*364)/(365*364^3) = 1/364$

• What does the presence of $S$ change to this problem?b – Marc van Leeuwen Jul 12 '12 at 16:18
• We may assume that $Q$ was born on December $31$, leaving the other $364$ days for the others. Whatever $R$'s birthday is, the probability $P$'s matches it is $1/364$. – André Nicolas Jul 12 '12 at 18:08

## 3 Answers

Two cases: If $P$ and $R$ share same birthday, the number of choices $=364$, otherwise, the number of choices $=2{364\choose 2}=364*363$. Thus, the probability that $P$ and $R$ same birthday $=\frac{364}{364+364*363}=\frac{364}{364^2}=\frac{1}{364}$

• While this is true, it is needlessly complicated. – Théophile Jul 12 '12 at 17:02
• i agree with you, but it's not so complicated and above that it's better to have solutions approached with different thoughts. – Aang Jul 12 '12 at 17:06
• Sure, different approaches are good. I'd say that if you're going to involve ${364\choose 2}$, though, then it would be clearer to keep it in factorized form as $\frac{(364)(363)}{2}$ rather than multiplying it out. At a glance, it isn't obvious—at least not to me—that $132496 = 364^2$. – Théophile Jul 12 '12 at 18:18
• I too got $1/364$ but my teacher says its wrong. I guess he's wrong this time. – Bazinga Jul 13 '12 at 2:53

If Q's birthday is different from everyone else's, then there are 364 choices for P and R. Thus, the probability that they are the same is indeed 1/364; the calculation need not be more complicated than that.

The way I see it is to fix P's birthday, and consider R. Since we know that neither of them share a birthday with Q, there are 364 different possibilities for R, each with equal probability. One of those possibilities is P's birthday, and thus the probability is 1/364.

Alternatively, you could just note that since neither share a birthday with Q, there are 364^2 ways to choose birthdays for P and R, 364 of which result in the two of them having the same birthday. Thus, the probability is 364/(364^2) = 1/364.