Question 1. IThis question is based on Problem 1, p. 172 int Laura chooses two b

Question

Question 1. IThis question is based on Problem 1, p. 172 int Laura chooses two balls at 2 in the textbook.] dom trom an urn containing 2 white, 4 black, and 8 orange at Laura wins $2 for each black ball selected, and she loses S1 for any balls. Suppose th t orange ball selected. Let X be the amount of money Laura wins. (a) Show that the possible values of X are 4, 1, andl -2, and calculate the corresponding probabilities. (b) Prove that P(X 0)>1/2 (c) Prove that E(X) =-2/7 and E(X") =316/91. (d) Ludo's Casino offers this game to the public, charging $1 for each play of the game. I Laura plays the game 70 times then what is her

Explanation / Answer

Question 1

There are 2 white, 4 black and 8 orange balls.

Black ball will give = $ 2

Other color will take = $1

Laura can choose only 2 balls so they can be 2 black balls, 1 black-1 other color ball or 2 other color balls.

when they are of black then total earning = 2 * $ 2 = $ 4

when one of them is black and other one is of other color, total earning = $ 2 - $ 1 = $1

When both of the balls are from other color, total earning = (-$1) * 2 = -$2

(b) Here probability of getting two black balls are or X = 4 is = Pr(2 black balls out of 4)/ Pr(2 balls out of 14)

= 4C2 / 14C2 = 4 * 3/(14 * 13) = 6/ 91

Probability of getting one black ball and one other color ball = 4C1 * 10C1/14C2  = 40/ 91

Pr(getting two other color balls) = 10C2/14C2 = (10 * 9)/(14 * 13) = 45/46

so total probability that Pr(X >0) = 6/91 + 40/91 = 46/91

(c) E(X) = xp(X) =  4 * 6/91 + 1 * 40/91 - 2 * 45/91 = -2/7

E(X2) = x2p(X) = 42 * 6/91 + 12 * 40/91 + 22 * 45/91 = 316/91

(d) In one game, her expected loss = 2/7

so in 70 games, her expected loss = 70 * 2/7= $ 20

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