Maths Olympiad Prep

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Combinatorics Difficulty 5.0 AIME Prove it United States

Problem:

Two jokers are added to a 5252 card deck and the entire stack of 5454 cards is shuffled randomly. What is the expected number of cards that will be between the two jokers?

Solution

Solution:

Each card has an equal likelihood of being either on top of the jokers, in between them, or below the jokers. Thus, on average, 1/31 / 3 of them will land between the two jokers.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.