Question

An urn contains one white chip and a second chip that is equally likely to be white or black. A chip is drawn at random and returned to the urn. Then a second chip is drawn.What is the probability that a white appears on the second draw given that a white appeared on the first draw? (Hint: Let Wi be the event that a white chip is selected on the ith draw, i = 1, 2. Then P(W2|W1) = P(W1∩W2) P(W1) . If both chips in the urn are white, P(W1) = 1; otherwise, P(W1) = 1/2)

Answer 1

Answer:

$P(W_{2} | W_{1})$ = 1 if both chips are white, $P(W_{2} | W_{1})$ = 1/2 otherwise

Step-by-step explanation:

$W_{i}$ represents the event that a white chip is selected on the ith draw. Then $P(W_{2} | W_{1}) = P(W_{2}\cap W_{1})/P(W_{1})$. If both chips in the urn are white, $P(W_{2}\cap W_{1}) = 1$ and $P(W_{1}) = 1$, then $P(W_{2} | W_{1}) = P(W_{2}\cap W_{1})/P(W_{1}) = 1$. If one chip is white and the other is black, then $P(W_{2} | W_{1}) = P(W_{2}\cap W_{1})/P(W_{1}) = P(W_{2})P(W_{1})/P(W_{1}) = P(W_{2}) = 1/2$ because the event a white chip is selected on the first draw is independent from the event  a white chip is selected on the second draw, and because the chips are drawn at random.