If there is a room of 400 people, what is the probability that at least two people share...
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If there is a room of 400 people, what is the probability that at least two people share the exact same birthday (assuming 365 or 366 days in a year)?
Show answer & explanation
With 400 people and only 365 possible birthdays, the pigeonhole principle guarantees that at least two people must share the same birthday. By the pigeonhole principle, if you have more items (400 people) than containers (365 days), at least one container must hold more than one item. Therefore, the probability is exactly 100%, not approximate.
Step-by-step Derivation:
Using the pigeonhole principle: We have 400 people and at most 365 possible distinct birthdays (ignoring leap years for standard calculation, or 366 if accounting for leap years). Since 400 > 365, we cannot assign each person a unique birthday. By the pigeonhole principle, at least two people must share the same birthday with absolute certainty. Therefore, P(at least two share a birthday) = 1.0 = 100%.