# Two Jokers problem

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.

• 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. Sep 9, 2020 at 19:17
• 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.
– lulu
Sep 9, 2020 at 19:29

$$\def\one{{\bf 1}}\def\ex{{\bf E}}\def\pr{{\bf P}}$$ 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 $$\ex\{N\} = \pr\{A_1\} + \cdots + \pr\{A_n\} = {52\over 3}.$$