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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.
Solution
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` |
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