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Suppose that I am in a casino. The dealer is known to deal cards from a deck that is comprised of 5 standard card decks. This means that there are 20 of each rank of card and 260 total cards. The dealer plays a game where he shuffles all 5 decks together randomly, and then he starts turning over cards from the top of the deck. Once he turns over 5 kings, the game is over. You win the game if you are closest to how many cards total (including the final king) the dealer turns over before the game ends. How many cards should you guess are turned over if you want to maximize your winning chances?

My reasoning The 20 kings partition the deck into 21 slices - something like ccccKccccK...ccccKccccKccc, where c denotes an arbitrary card, and K denotes a king. Each slice has expected size 260/21, and so for 5 kings to occur, there must be 1300/21 cards expected. The correct answer is 1305/21, however. Where am I going wrong? Moreover, how can I be more rigorous here? Can we use the idea of spacings somehow?

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  • $\begingroup$ Hint: Working from the back of the deck, this is equivalent to finding the expected place of the first King. $\endgroup$
    – lulu
    Commented May 8, 2022 at 20:09
  • $\begingroup$ @lulu there are $20$ kings in the five decks. $\endgroup$
    – Thomas Andrews
    Commented May 8, 2022 at 20:09
  • $\begingroup$ @ThomasAndrews Oh, right. $\endgroup$
    – lulu
    Commented May 8, 2022 at 20:10
  • $\begingroup$ In that case, use symmetry. Suppose we add a single Joker. Now imagine the cards arranged in a circle. By Symmetry, we expect a constant gap between the $21$ cards (Kings and the Joker). Compute that gap. Now remove the Joker to define the first and last cards of the shuffle. $\endgroup$
    – lulu
    Commented May 8, 2022 at 20:11

1 Answer 1

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Partitioning the deck into slices is almost right idea, but you need to exclude the kings from the count. This is because only twenty of the twenty-one slices contain a king in your method, so the slices do not all have the same expected size in your counting scheme.

There are $240$ non-kings, so the expected number of non-kings in each slice is $240/21$. This means the expected number of cards dealt is $5\times (240/21)+5$, where the $+5$ accounts for the five kings which come after the first five slices.

By the way, the question asks for "maximizing your winning chances", but in your solution you compute the "expected number of cards dealt". I see no reason for these to be the same concept, and would expect that that you would actually maximize your chances by choosing the mode, not the mean.

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