Question

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

Answer 1

Let X denote the number of aces among the two cards drawn with replacement. Clearly, 0.1 and 2 are the possible values of X since the draws are with replacement, the outcomes of the two draws are independent of each other. Also. since there are 4 aces in the deck of 52 cards, P (an ace) = `4/52 = 1/13`, and P[a non-ace] = `12/13`.

Then P[X = 0] = P[non-ace and non-ace]

 = `12/13 xx 12/13`

= `144/169`

P[X = 1] = P[ace and non-ace] + P[non-ace and ace]

= `1/13 xx 12/13 + 12/13 xx 1/13`

= `24/169`

and P[X = 2] = P[ace and ace]

= `1/13 xx 1/13`

= `1/169`

The required probability distribution is then as follows:

x	0	1	2
P[X = x]	`144/169`	`24/169`	`1/169`