Box P has 2 red balls and 3 blue balls. Box Q has 3 red balls and 1 blue ball.…
2005
Box P has 2 red balls and 3 blue balls. Box Q has 3 red balls and 1 blue ball. A ball is selected as follows:
(i) Select a box.
(ii) Choose a ball from the selected box, with each ball in that box equally likely to be chosen.
The probabilities of selecting boxes P and Q are 1/3 and 2/3, respectively. Given that the selected ball is red, what is the probability that it came from box P?
- A.
4/19
- B.
5/19
- C.
2/9
- D.
19/30
Attempted by 4 students.
Show answer & explanation
Correct answer: A
Let R be the event that the selected ball is red.
P(P) = 1/3 and P(Q) = 2/3.
P(R | P) = 2/5, because box P has 2 red balls out of 5.
P(R | Q) = 3/4, because box Q has 3 red balls out of 4.
By Bayes theorem,
P(P | R) = [P(P)P(R | P)] / [P(P)P(R | P) + P(Q)P(R | Q)]
= [(1/3)(2/5)] / [(1/3)(2/5) + (2/3)(3/4)]
= (2/15) / (2/15 + 1/2)
= (2/15) / (19/30)
= 4/19.