For independent uniform draws from labels, the probability of all labels being distinct is when , and zero otherwise. Applying gives a collision probability at least . The transition therefore occurs at sample size of order , not : many pairs can collide before many labels have been sampled.
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The Birthday Problem, also known as the Birthday Paradox, refers to a counterintuitive probability puzzle that deals with the likelihood of two or more people sharing the same birthday in a group. The problem is commonly stated as follows: In a group of \( n \) people, what is the probability that at least two of them share the same birthday?