Exercise 5.4.4

$\star$ How many people should be invited to a party in order to make it likely that there are three people with the same birthday?

The answer is $88$. I reached it by trial and error. But let's analyze it with indicator random variables.

Let $X_{ijk}$ be the indicator random variable for the event of the people with indices $i$, $j$ and $k$ have the same birthday. The probability is $1/n^2$. Then:

$$ \begin{align} \E[X] &= \sum_{i=1}^n\sum_{j=i+1}^n\sum_{k=j+1}^nX_{ijk} \\ &= \sum_{i=1}^n\sum_{j=i+1}^n\sum_{k=j+1}^n\frac{1}{n^2} \\ &= \binom{n}{3}\frac{1}{n^2} \\ &= \frac{k(k-1)(k-2)}{6n^2} \end{align} $$

Solving this yields $94$. It's a bit more, but again, indicator random variables are approximate.

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