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Nov 2023 p53 q3
2582
Tim has two bags of marbles, A and B.
Bag A contains 8 white, 4 red and 3 yellow marbles.
Bag B contains 6 white, 7 red and 2 yellow marbles.
Tim also has an ordinary fair 6-sided dice. He rolls the dice. If he obtains a 1 or 2, he chooses two marbles at random from bag A, without replacement. If he obtains a 3, 4, 5 or 6, he chooses two marbles at random from bag B, without replacement.
Find the probability that the two marbles come from bag B given that one is white and one is red.
Solution
To find the probability that the two marbles come from bag B given that one is white and one is red, we use conditional probability:
\(P(B \mid WR \text{ or } RW) = \frac{P(W \& R \text{ from bag B})}{P(W \text{ and } R)}\)
First, calculate \(P(W \& R \text{ from bag B})\):
\(P(W \& R \text{ from bag B}) = \frac{2}{3} \times \frac{6}{15} \times \frac{7}{14} + \frac{2}{3} \times \frac{7}{15} \times \frac{6}{14} = \frac{4}{15}\)
Next, calculate \(P(W \text{ and } R)\):
\(P(W \text{ and } R) = P(W \& R \text{ from bag A}) + P(W \& R \text{ from bag B})\)
\(P(W \& R \text{ from bag A}) = \frac{2}{6} \times \frac{8}{15} \times \frac{4}{14} + \frac{2}{6} \times \frac{4}{15} \times \frac{8}{14} = \frac{116}{315}\)
