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Question
Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of kings.
Solution
Let X denote the number of kings in a sample of 2 cards drawn from a well-shuffled pack of 52 playing cards. Then, X can take the values 0, 1 and 2.
Now,
\[P\left( X = 0 \right)\]
\[ = P\left( \text{ no kings } \right)\]
\[ = \frac{48}{52} \times \frac{48}{52}\]
\[ = \frac{12 \times 12}{13 \times 13}\]
\[ = \frac{144}{169}\]
\[P\left( X = 1 \right)\]
\[ = P\left( 1 \text{ king } \right)\]
\[ = \frac{4}{52} \times \frac{48}{52}\]
\[ = \frac{2 \times 12}{13 \times 13}\]
\[ = \frac{24}{169}\]
\[P\left( X = 2 \right)\]
\[ = P\left( 2 \text{ kings } \right)\]
\[ = \frac{4}{52} \times \frac{4}{52}\]
\[ = \frac{1 \times 1}{13 \times 13}\]
\[ = \frac{1}{169}\]
Thus, the probability distribution of X is given by
| X | P(X) |
| 0 |
\[\frac{144}{169}\]
|
| 1 |
\[\frac{24}{169}\]
|
| 2 |
\[\frac{1}{169}\]
|
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