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Two jokers are added to a $52$ card deck and the entire stack of $54$ cards is shuffled randomly. What is the expected number of cards that will be strictly between the two jokers?

This is an HMMT problem that I found online and wasn't sure how to solve.

I tried $\frac{37}{2}$ but that was incorrect.

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  • $\begingroup$ For the problem "2 jokers are added to an $n$-card deck", check small values of $n$ and see if you can find a pattern. $\endgroup$
    – player3236
    Sep 9, 2020 at 19:17
  • $\begingroup$ As another approach: For each of the $52$ cards, consider the indicator variable which tells you if the given card falls between the two jokers. Now use linearity. $\endgroup$
    – lulu
    Sep 9, 2020 at 19:29

2 Answers 2

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$\def\one{{\bf 1}}\def\ex{{\bf E}}\def\pr{{\bf P}}$ Recall that an indicator variable for an event $A$ is the function $$\one_A = \cases{1, & if $A$ occurs;\cr 0, & if $A$ does not occur.}$$ Following the approach outlined by lulu in the comment above, we start by numbering the 52 non-Joker cards $C_1,\ldots,C_{52}$. For each $i\in\{1,\ldots,52\}$, let $A_i$ denote the event "card $C_i$ falls between both Jokers". Let $N$ be the number of cards that fall between the two Jokers. We have $$\eqalign{ \ex\{N\} &= \ex\{\one_{A_1} + \cdots + \one_{A_n}\} \cr &= \ex\{\one_{A_1}\} + \cdots + \ex\{\one_{A_n}\} \cr &= \pr\{A_1\} + \cdots + \pr\{A_n\}. \cr }$$ For each $i$, the probability that card $C_i$ comes between both Jokers $J_1$ and $J_2$ is $1/3$, since in the shuffle of all 54 cards, each of the six sub-permutations of $\{J_1, C_i, J_2\}$ are equally likely, and two of those six have $C_i$ in the middle. This is true for all $i$, so $$\ex\{N\} = \pr\{A_1\} + \cdots + \pr\{A_n\} = {52\over 3}.$$

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Imagine the 2 jokers placed on the table between 3 bins A, B, C. The other 52 cards will be dealt at random into bin A, B, or C. On average there will be 52/3 cards in each bin. The expected number of cards between the 2 jokers is the number in bin B, which is 52/3.

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