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# Difficult Expected Value Problem

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A standard deck of cards contains 52 cards. These 52 cards are arranged in a circle, at random. Find the expected number of pairs of adjacents cards that are both

I. Aces.

II. face cards (King, Queen, and Jack).

Oct 18, 2023

#1
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i.

There are 52 cards in a deck, so there are 51 pairs of adjacent cards. Each pair has an equal probability of being aces, which is 4/52 * 3/51 = 12/221. So, the expected number of pairs of adjacent cards that are both aces is:

Expected number = 51 * (12/221) = 40/221

Therefore, the expected number of pairs of adjacent cards that are both aces is 40/221.

ii.

There are 12 face cards in a deck, so there are 11 pairs of adjacent face cards. Each pair has an equal probability of being face cards, which is 12/52 * 11/51 = 22/85. So, the expected number of pairs of adjacent cards that are both face cards is:

Expected number = 11 * (22/85) = 2/5

Therefore, the expected number of pairs of adjacent cards that are both face cards is 2/5.

Oct 27, 2023
#2
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The answers turned out to be 4/17 for part I and 44/17 for part II on the answers sheet. Thanks for the effort, bingboy, and I don't mean to be confrontational, but your arithmetic seems to be completely wrong. I'm pretty sure that 4/52 * 3/51 = 1/221, and 51 * (12/221) = 36/13. 11 * (22/85) = 242/85, too.

knotW  Oct 27, 2023
edited by knotW  Oct 28, 2023