Question

Suppose a box contains one blue card and six red cards. Each of the six

red cards has a unique label from {A, B, C, D, E, F}. Suppose three of these cards are selected randomly, without replacement.
(a) If it is known that the card labeled A has been selected, what is the probability that all three cards are red?
(b) If it is known that at least one red card has been selected, what is the probability that all three cards are red?

Answer 1

The probability that all three cards are red, given that Card A has been selected, is 10/21. The probability that all three cards are red, given that at least one red card has been selected, is 5/3.

Step-by-step explanation:

To solve the problem, we need to use conditional probability.

(a) The probability that all three cards are red, given that the card labeled A has been selected, is determined by the probability of selecting two additional red cards out of the remaining five red cards.

P(All 3 cards are red | Card A selected) = P(Selecting 2 red cards out of remaining 5 red cards)

P(All 3 cards are red | Card A selected) = (5/7) * (4/6) = 20/42 = 10/21

(b) The probability that all three cards are red, given that at least one red card has been selected, is determined by the probability of selecting two additional red cards out of the remaining five red cards, divided by the probability of selecting at least one red card.

P(All 3 cards are red | At least one red card selected) = P(Selecting 2 red cards out of remaining 5 red cards) / P(At least one red card selected)

P(All 3 cards are red | At least one red card selected) = (5/7) * (4/6) / (1 - P(Selecting no red cards))

P(All 3 cards are red | At least one red card selected) = (5/7) * (4/6) / (1 - (6/7) * (5/6)) = 20/42 / (1 - 30/42) = 20/42 / 12/42 = 20/12 = 5/3