Paper 2, Section I, 3F

Probability | Part IA, 2019

(a) Prove that log(n!)nlogn\log (n !) \sim n \log n as nn \rightarrow \infty.

(b) State Stirling's approximation for nn !.

(c) A school party of nn boys and nn girls travel on a red bus and a green bus. Each bus can hold nn children. The children are distributed at random between the buses.

Let AnA_{n} be the event that the boys all travel on the red bus and the girls all travel on the green bus. Show that

P(An)πn4n as n\mathbb{P}\left(A_{n}\right) \sim \frac{\sqrt{\pi n}}{4^{n}} \text { as } n \rightarrow \infty

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