two cards are drawn successively with replacement from a pack of 52 cards. Find the probability distribution of the number of diamonds and the mean of the distribution.

Explanation

There are `13` diamonds in a pack of `52` cards.

So, probability when both drawn cards are not diamond, `P(D =0) = (52-13)/52**(52-13)/52 = 39/52*39/52 = 9/16`

Probabilty of drawing `1` diamond in any of the draw, `P(D =1) = (13/52**39/52)+(39/52**13/52) = 3/8`

Probabilty of drawing `2` diamonds, `P(D =2) = (13/52**13/52) = 1/16`

Now, mean of the given distribution can be given as,

`M = sum_(i=1)^n P_ix_i`

`M = 0*9/16+1*3/8+2*1/16 = 3/8+1/8 = 1/2`

So, mean of the distribution is `1/2.`