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Q.

Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. If the variance of the number of kings is $σ^{2}$ then ${(221)}^{2} σ^{2}$ is__

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answer is 6800.

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Detailed Solution

Let X denote the number of kings in a draw of two cards

X	0	1	2
$P (X)$	$\frac{^{48} C_{2}}{^{52} C_{2}} = \frac{188}{121}$	$\frac{^{4} {C_{1}}^{48} C_{1}}{^{52} C_{2}} = \frac{32}{221}$	$\frac{^{4} C_{2}}{^{52} C_{2}} = \frac{1}{221}$

Now, Mean of $X = \sum_{i = 1}^{n} x_{i} p (x_{i}) = \frac{34}{221}$
Also, $E (X^{2}) = \sum_{i = 1}^{n} x_{i}^{2} p (x_{i}) = \frac{36}{221}$
This, $var (x) = E (X^{2}) - {(E (X))}^{2}$

$= \frac{36}{221} - {(\frac{34}{221})}^{2} = \frac{6800}{{(221)}^{2}}$

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