Two cards are drawn successively with replacement from a well-shuffled

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Find the probability distribution of the number of aces.

Answer»

The number of aces is a random variable. Let it be denoted by X. Clearly, X can take the values 0, 1, or 2.

Now, since the draws are done with replacement, therefore, the two draws form independent experiments.

Therefore, P(X = 0) = P(non-ace and non-ace) = P(non-ace) × P(non-ace) = 48/52 x 48/52 = 144/169

P(X = 1) = 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)

= 4/52 x 48/52 + 48/52 x 4/52 = 24/169

and P(X = 2) = P (ace and ace) = 4/52 x 4/52 = 1/169

Thus, the required probability distribution is

X	0	1	2
P(X)	144/169	24/169	1/169

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