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Question
Two cards are drawn simultaneously from a well-shuffled deck of 52 cards. Find the probability distribution of the number of successes, when getting a spade is considered a success.
Solution
Let X denote the number of spades 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,
\[P\left( X = 0 \right)\]
\[ = P\left( \text{ no spade} \right)\]
\[ = \frac{{}^{39} C_2}{{}^{52} C_2}\]
\[ = \frac{741}{1326}\]
\[ = \frac{19}{34}\]
\[P\left( X = 1 \right)\]
\[ = P\left( 1 \text{ spade } \right)\]
\[ = \frac{{}^{13} C_1 \times^{39} C_1}{{}^{52} C_2}\]
\[ = \frac{507}{1326}\]
\[ = \frac{13}{34}\]
\[P\left( X = 2 \right)\]
\[ = P\left( 2 \text{ spades } \right)\]
\[ = \frac{{}^{13} C_2}{{}^{52} C_2}\]
\[ = \frac{78}{1326}\]
\[ = \frac{1}{17}\]
Thus, the probability distribution of X is given by
| X | P(X) |
| 0 |
\[\frac{19}{34}\]
|
| 1 |
\[\frac{13}{34}\]
|
| 2 |
\[\frac{1}{17}\]
|
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