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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?

 Mar 31, 2019

\(\text{All pairs of positions }(m,n) \text{ of the jokers are equally likely}\\ P[0]=P[|m-n|=1] =\dfrac{53}{\dbinom{54}{2}}=\dfrac{2}{27}\\ P[1] = P[|m-n|=2] = \dfrac{52}{\dbinom{54}{2}} = \dfrac{104}{1431}\\ P[k] = P|m-n|=k+1 = \dfrac{54-k-1}{\dbinom{54}{2}}\)


\(\text{The expected value of the number of cards between jokers is}\\ \sum \limits_{k=1}^{52}\dfrac{54-k-1}{\dbinom{54}{2}}k = \\ \dfrac{1}{1431}\left(\sum \limits_{k=1}^{52} 53k-k^2 \right) = \\ \dfrac{1}{1431}\left(\dfrac{53}{2}(52)(53) - \dfrac 1 6 (52)(53)(105) \right) = \dfrac{52}{3}\)

 Mar 31, 2019

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