Prove the law of total probability: if $A_{1}, \ldots, A_{n}$ are pairwise disjoint events with $\mathbb{P}\left(A_{i}\right)>0$, and $B \subseteq A_{1} \cup \ldots \cup A_{n}$ then $\mathbb{P}(B)=\sum_{i=1}^{n} \mathbb{P}\left(A_{i}\right) \mathbb{P}\left(B \mid A_{i}\right)$.

There are $n$ people in a lecture room. Their birthdays are independent random variables, and each person's birthday is equally likely to be any of the 365 days of the year. By using the bound $1-x \leqslant e^{-x}$ for $0 \leqslant x \leqslant 1$, prove that if $n \geqslant 29$ then the probability that at least two people have the same birthday is at least $2 / 3$.

[In calculations, you may take $\sqrt{1+8 \times 365 \ln 3}=56.6$.]