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Three cards are drawn successively with replacement from a well-shuffled deck of 52 cards. A random variable X denotes the number of hearts in the three cards drawn. determine the probability distribution of X. Select the correct answer from above options

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Correct Answer - Mean = 3 4 ,variance = 9 16 There are 13 hearts and 39 other cards. Let E= event of drawing a heart. Then, P(E)= 13 52 = 1 4 andP( ˉ E )=(1- 1 4 )= 3 4 . Let X= number of hearts in a draw. Then, X=0,1,2 or 3. P(X=0)=P( ˉ E ˉ E ˉ E )=P( ˉ E )×P( ˉ E )×P( ˉ E )=( 3 4 × 3 4 × 3 4 )= 27 64 . P(X=1)=P[(E ˉ E ˉ E )or( ˉ E ˉ E E)] =P(E ˉ E ˉ E )+P( ˉ E E ˉ E )+P( ˉ E ˉ E E) =( 1 4 × 3 4 × 3 4 )+( 3 4 × 1 4 × 3 4 )+( 3 4 × 3 4 × 1 4 )= 27 64 . P(X=2)=P[(EE ˉ E )or(E ˉ E E)or( ˉ E EE)] =P(EE ˉ E )+P(E ˉ E E)+P( ˉ E EE) =( 1 4 × 1 4 × 3 4 )+( 1 4 × 3 4 × 1 4 )+( 3 4 × 1 4 × 1 4 )= 9 64 . P(X=3)=P(EEE)=P(E)×P(E)×P(E)=( 1 4 × 1 4 × 1 4 )= 1 64 . Find the mean and variance.

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