Paper 2, Section II, F

Probability | Part IA, 2009

Prove the law of total probability: if A1,,AnA_{1}, \ldots, A_{n} are pairwise disjoint events with P(Ai)>0\mathbb{P}\left(A_{i}\right)>0, and BA1AnB \subseteq A_{1} \cup \ldots \cup A_{n} then P(B)=i=1nP(Ai)P(BAi)\mathbb{P}(B)=\sum_{i=1}^{n} \mathbb{P}\left(A_{i}\right) \mathbb{P}\left(B \mid A_{i}\right).

There are nn 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 1xex1-x \leqslant e^{-x} for 0x10 \leqslant x \leqslant 1, prove that if n29n \geqslant 29 then the probability that at least two people have the same birthday is at least 2/32 / 3.

[In calculations, you may take 1+8×365ln3=56.6\sqrt{1+8 \times 365 \ln 3}=56.6.]

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