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

 

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

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 Mar 31, 2019

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