Question

this is the last problem. 6. A jar contains 5 red balls and 5 black balls. Two balls are successively drawn from the jar. After the first draw the color of the ball is noted, and then the selected ball is returned to the jar together with another ball of the opposite color. In other words, if a red ball is drawn it is returned to the jar together with another black ball; if black ball is drawn it is returned to the jar together with another red ball. Assume that for both draws all the balls currently in the jar are equally likely to be selected. Let R1, R2 be the events, that on draws 1 and 2, respectively, the selected ball is red. Let B1, B2 be the events, that on draws 1 and 2, respectively, the selected ball is black

7. Referring to the last problem, suppose that one receives $1 for each red ball drawn, and pays $1 for each black ball drawn. Let the random variable W be the amount won on the two draws. ( Note a loss is considered a negative amount won. )

(a) Find P( W = w ) for all relevant values of w.

(b) Find E[W].

(c) Find Var[W].

Answer 1

W can take values $-2, $0 and $2

W = $-2 if both balls drawn are black

W = $0 if one of the balls drawn is red and one is black

W = $2 if both balls drawn are red

(a)

P(W = -2) = 5/10*5/11 = 5/22

P(W = 0) = 5/10*6/11 + 5/10*6/11 = 6/11

P(W = 2) = 5/10*5/11 = 5/22

(b)

E[W] = -2*5/22 + 0*6/11 + 2*5/22 = 0

(c)

Var[W] = = 20/11 = 1.82