Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then P(X = 1) + P (X = 2) equals :
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Official solution
P (X = 1) means out of two drawn cards one card is ace.
and P(X = 2) means both the drawn cards are ace.
\[ \therefore \] P(X = 1) = first card is ace or 2nd card is ace.
= A \_ + \_ A
= \[{4 \over {52}} \times {{48} \over {52}} + {{48} \over {52}} \times {4 \over {52}}\]
= \[2 \times {4 \over {52}} \times {{48} \over {52}}\]
P(X = 2) = First and second both cards arc ace.
= A A
= \[{4 \over {52}} \times {4 \over {52}}\]
\[ \therefore \] P(X = 1) + P(X = 2)
= \[2 \times {4 \over {52}} \times {{48} \over {52}} + {4 \over {52}} \times {4 \over {52}}\]
= \[{{25} \over {169}}\]