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.

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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,

$P (X = 0) = P (no spade) = \frac{^{39} C_{2}}{^{52} C_{2}} = \frac{741}{1326} = \frac{19}{34} P (X = 1) = P (1 spade) = \frac{^{13} C_{1} \times^{39} C_{1}}{^{52} C_{2}} = \frac{507}{1326} = \frac{13}{34} P (X = 2) = P (2 spades) = \frac{^{13} C_{2}}{^{52} C_{2}} = \frac{78}{1326} = \frac{1}{17}$

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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