In Between

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Hayley has a deck of 104 cards, each numbered distinctly with an integer k k , where 1 ≤ k ≤ 104 1 \leq k \leq 104 . Hayley randomly picks 2 cards, with the numbers k 1 k_1 and k 2 k_2 written on them, where k 2 > k 1 k_2>k_1 . Without replacing the first 2 cards, she then randomly picks a third card with the number k 3 k_3 written on it. What is the probability that k 1 < k 3 < k 2 k_1<k_3<k_2 ?


The answer is 0.3333.

Ryan Phua by Ryan Phua , 22, Singapore

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1 solution

Lionslayer Lim
Feb 8, 2014

No matter which 3 cards are picked, there is a 1 / 3 1/3 chance that each card ( k 1 k_1 , k 2 k_2 or k 3 k_3 ) has the middle value. Hence the probability that k 1 < k 3 < k 2 k_1<k_3<k_2 is 1 / 3 = 0.333 1/3=\boxed{0.333}

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