Question

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.

(a) Find P(R1)

(b) Find P(R2|R1)

(c) Find P(R2 ∩ R1)

(d) Find P(B2|R1)

(e) Find P(B2 ∩ R1)

(f) Find P(R2 ∩ B1)

(g) Find P(R2)

(h) Are the events R1 and R2 independent? Justify your answer using the definition of independence.

Answer 1

(a) P(R1) = 5/10 = 1/2

(b) P(R2 | R1) = 5/11 (Since one black ball is added if R1 occurs)

(c) P(R2 R1) = P(R2 | R1)*P(R1) = 5/11 * 1/2 = 5/22

(d) P(B2 | R1) = 6/11

(e) P(B2 R1) = P(B2 | R1)*P(R1) = 6/11 * 1/2 = 3/11

(f) P(R2 B1) = P(R2 | B1)*P(B1) = 5/11*1/2 = 5/22

(g) P(R2) = P(R2 R1) + P(R2 B1) = 5/22 + 5/22 = 5/11

(h) Here, P(R2 | R1) = P(R2)

Thus, R1 and R2 are independent events

(Since the probability of event R2 conditioned on R1 is equal to the unconditioned probability of R2)