Question

Two cards are drawn simultaneously (or successively without replacement) for a well shuffled of 52 cards. Find the mean, variance and standard deviation of the number of kings.

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Answer 1

$\huge \tt \blue{Question}$

Two cards are drawn simultaneously (or successively without replacement) for a well shuffled of 52 cards. Find the mean, variance and standard deviation of the number of kings.

$\huge \tt \blue{Solution}$

Let X denote the number of kings in a draw of two cards. X is a random variable which can assume the values 0, 1 or 2.

Now P(X = 0) = P (no king)

$= \frac{48 c2}{52c2}$

$= \frac{48 \times 57}{52 \times 51}$

$= \frac{188}{221}$

P ( X = 1 ) = P ( exactly one card is king )

$= \frac{4c2}{52c2}$

$= \frac{4 \times 3}{52 \times 51}$

$= \frac{1}{221}$

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$= ( {0}^{2} \times \frac{188}{221} + {1}^{2} \times \frac{32}{221} + {2}^{2} \times \frac{1}{221} ) - ( \frac{34}{221} ) ^{2}$

$= \frac{36}{221} - \frac{1156}{48841}$

$= 0.1392$

$σ \: = \sqrt{Var(X)}$

$= \sqrt{0.1392}$

$\pink { = 0.3730}$