People are arriving at a party one at a time. While waiting...

People are arriving at a party one at a time. While waiting for more people to arrive they entertain themselves by comparing their birthdays. Let $X$ be the number of people needed to obtain a birthday match, i.e., before person $X$ arrives no two people have the same birthday, but when person $X$ arrives there is a match. Find the PMF of $X$.

Key Concepts

Uniform Distribution and Independence

The assumption that each person’s birthday is drawn independently and uniformly from the set of possible days is critical to the analysis. This ensures that each birthday is equally likely, allowing the use of combinatorial formulas to calculate probabilities accurately.

Combinatorial Analysis

This concept involves counting strategies such as permutations and combinations to determine the number of ways events can occur. In the birthday problem, it is used to compute the probability of having a set of unique birthdays among a specific number of people and then incorporating the event of a duplicate occurring.

Birthday Problem

This is a classic probability puzzle that examines the counterintuitive probability of shared birthdays in a group. It involves calculating the likelihood that, in a set of randomly selected individuals, at least two of them share the same birthday, leading to surprising results even in relatively small groups.

Discrete Random Variable

A discrete random variable is one that takes on a countable number of distinct values. In the context of the problem, the number of people required to produce the first repeated birthday is a discrete random variable, and its distribution is defined by specific probabilities for each possible count.

Probability Mass Function (PMF)

The PMF specifies the probability that a discrete random variable assumes a particular value. For this problem, it describes the probability that the first duplicate birthday occurs exactly when the X-th person arrives, conditional on the successive outcomes for each arrival.

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