Two cards are drawn successively without replacement from a well shuffled deck of cards. Find the mean and standard variation of the random variable X where X is the number of aces.


Let X denotes a random variable of number of aces

Clearly, X can take values 0, 1 or 2 because only two cards are drawn.


Total deck of cards = 52


and total no. of ACE cards in a deck of cards = 4


Now, since the draws are done without replacement, therefore, the two draws are not independent.


Therefore,


P(X = 0) = Probability of no ace being drawn


= P(non – ace and non – ace)


= P(non – ace) × P(non – ace)




P(X = 1) = Probability that 1 card is an ace


= P(ace and non – ace or non –ace and ace)


= P(ace and non – ace) + P(non – ace and ace) = P(ace) P(non – ace) + P(non – ace) P(ace)





P(X = 2) = Probability that both cards are ace


= P(ace and ace)


= P(ace) × P(ace)




We know that,


Mean (μ) = E(X) = ΣXP(X)






Also, Var(X) = E(X2) – [E(X)]2


= ΣX2P(X) – [E(X)]2





= 0.1629 – 0.0237


= 0.1392


Standard Deviation = √Var(X) = √0.1392 0.373(approx.)

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