Question

A set of 20 cards consists of 12 red cards and 8 black cards. The cards are shuffled thoroughly and you choose one at random, observe its color, and replace it in the set. The cards are thoroughly reshuffled, and you again choose a card at random, observe its color, and replace it in the set. This is done a total of six times. Let X be the number of red cards observed in these six trials. The variance of X isA. 6.B. 3.60.C. 2.4.D. 1.44.

Answer 1

the answer D=1.44

Explanation:

if the X= number of red cards observed in 6 trials , since each card observation is independent from the others and the sampling process is done with replacement ( the card is observed, then returned and reshuffled) , X follows an binomial distribution.

X(x)= n!/((n-x)!*x!) *p^x *(1-p)^(n-x)

where n = number of trials = 6 , x= number of red cards observed , p= probability of obtaining a red card in one try

the probability of obtaining the card in one try is

p = number of red cards / total number of cards = 12/ (12+8) = 0.6

since we know that X has a binomial distribution, the variance of this kind of distribution is

variance = σ² = n * p * (1-p)

therefore the variance of X is

variance = σ² = n * p * (1-p) = 6 * 0.6 * (1-0.6) = 1.44