Question

Bag \( B_1 \) contains 6 white and 4 blue balls, Bag \( B_2 \) contains 4 white and 6 blue balls, and Bag \( B_3 \) contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability that the ball is drawn from Bag \( B_2 \) is:

• A. \( \frac{1}{3} \)
• B. \( \frac{2}{3} \)
• C. \( \frac{4}{15} \)
• D. \( \frac{2}{5} \)

Hint:

Use Bayes' Theorem for conditional probability when dealing with multiple events.

Answer 1

The Correct Option is C

To determine the probability that the ball originated from Bag \( B_2 \), given that a white ball was drawn, we apply Bayes' theorem. The events are defined as:

• \( A_1 \): Selection of Bag \( B_1 \)
• \( A_2 \): Selection of Bag \( B_2 \)
• \( A_3 \): Selection of Bag \( B_3 \)
• \( W \): Drawing a white ball

Our objective is to calculate \( P(A_2 \mid W) \), the posterior probability of the ball being from Bag \( B_2 \) given it is white. Bayes' theorem states:

\( P(A_2 \mid W) = \frac{P(W \mid A_2)P(A_2)}{P(W)} \)

Since each bag has an equal probability of selection:

\( P(A_1) = P(A_2) = P(A_3) = \frac{1}{3} \)

The conditional probabilities of drawing a white ball from each bag are:

\( P(W \mid A_1) = \frac{6}{10} = \frac{3}{5} \)

\( P(W \mid A_2) = \frac{4}{10} = \frac{2}{5} \)

\( P(W \mid A_3) = \frac{5}{10} = \frac{1}{2} \)

The total probability of drawing a white ball, \( P(W) \), is computed using the law of total probability:

\( P(W) = P(W \mid A_1)P(A_1) + P(W \mid A_2)P(A_2) + P(W \mid A_3)P(A_3) \)

\( P(W) = \frac{3}{5}\cdot\frac{1}{3} + \frac{2}{5}\cdot\frac{1}{3} + \frac{1}{2}\cdot\frac{1}{3} \)

\( P(W) = \frac{1}{5} + \frac{2}{15} + \frac{1}{6} \)

To sum these fractions, we find a common denominator of 30:

• \( \frac{1}{5} = \frac{6}{30} \)
• \( \frac{2}{15} = \frac{4}{30} \)
• \( \frac{1}{6} = \frac{5}{30} \)

Therefore:

\( P(W) = \frac{6}{30} + \frac{4}{30} + \frac{5}{30} = \frac{15}{30} = \frac{1}{2} \)

Substituting these values into Bayes' theorem for \( P(A_2 \mid W) \):

\( P(A_2 \mid W) = \frac{\frac{2}{5}\cdot\frac{1}{3}}{\frac{1}{2}} \)

\( P(A_2 \mid W) = \frac{\frac{2}{15}}{\frac{1}{2}} = \frac{2}{15} \cdot 2 = \frac{4}{15} \)

The probability that the white ball was drawn from Bag \( B_2 \) is \( \frac{4}{15} \).