086概率简单数值题short

The Birthday Paradox

题目

Assume birthdays are uniformly distributed over 365 days (ignore leap years). (a) What is the exact probability that in a room of 23 people, at least two share a birthday? (b) Derive a simple approximation for the number of people $n$ needed so that the probability of a shared birthday exceeds $1/2$ . (c) A trading desk has 50 traders. A manager claims that having two traders share a birthday is 'a remarkable coincidence.' Is the manager correct? Compute the probability and comment.

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c_probability

c_manager_correct

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Generalize: in a hash table with $d$ slots and $n$ random keys, what is the approximate probability of at least one collision? At what $n$ does this exceed $1/2$ ? How does this connect to the birthday attack in cryptography?

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