Q. 16

# Two cards are drawn without replacement from a well-shuffled deck of 52 cards. Let X be the number of face cards drawn. Find the mean and variance of X.

Answer :

Given : Two cards are drawn without replacement from a well-shuffled deck of 52 cards.

To find : mean (𝓊) and variance (σ^{2}) of X

Formula used :

Mean = E(X) =

Variance = E(X^{2}) -

Mean = E(X) = = x_{1}P(x_{1}) + x_{2}P(x_{2}) + x_{3}P(x_{3})

Two cards are drawn without replacement from a well-shuffled deck of 52 cards.

Let X denote the number of face cards drawn

There are 12 face cards present in 52 cards

P(0) = = =

P(1) = = =

P(2) = = =

The probability distribution table is as follows,

Mean = E(X) = 0() + 1() + 2() = 0 + + = = =

Mean = E(X) =

= =

E(X^{2}) = = P(x_{1}) + P(x_{2}) + P(x_{3})

E(X^{2}) = () + () + () = 0 + + =

E(X^{2}) =

Variance = E(X^{2}) - = – = =

Variance = E(X^{2}) - =

Mean = E(X) =

Variance =

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