Question

Two cards are drawn at random from an ordinary deck of 52 playing cards. If the two cards display the same suit, you win $2. If they are the same color only, you win $1. Otherwise, you lose 50 cent. Calculate

(a) the expected value of the amount you win;

(b) the variance of the amount you win;

Answer 1

Here let say x is the winnings.

P(Two cards of same suit) = P(First card is of a particular suit) * P(Second card is of particular suit) * Number of suit

= 4 * (13/52) * (12/51) = 0.2353

P(Two cards of same color) = P(First card is of a pariticular suit) * P(Second card is of a pariticular color but different suit) * Number of Suit

= 13/52 * 13/51 * 2 = 0.2549

P(When there is no such case) = 1 - 0.2353 - 0.2449 = 0.5098

(a) p(x) = 0.2353 ; x = $ 2

= 0.2549; x = $ 1

= 0.5098; x = - $ 0.5

E[x] = 0.2353 * 2 + 0.2549 * 1 + 0.5098 * (-0.5) = $ 0.4706

(b) VaR[x] = E[x2] - E[x]2

E[x2] = 0.2353 * 2 * 2 + 0.2549 * 1 * 1 + 0.5098 * (-0.5) * (-0.5) = 1.3256

VaR[x] = 1.3256 - 0.47062 = 1.1021