To find the probability that the chosen ball came from urn B given that it was a yellow ball, we can use Bayes' theorem:
\[ P(B|Y) = \frac{P(Y|B) \times P(B)}{P(Y)} \]
Where:
- \( P(B|Y) \) is the probability that the ball came from urn B given that it was yellow.
- \( P(Y|B) \) is the probability of picking a yellow ball given that the ball came from urn B. (Given)
- \( P(B) \) is the probability of picking urn B.
- \( P(Y) \) is the probability of picking a yellow ball.
Given:
- \( P(Y|B) = \frac{5}{17} \) (Probability of picking a yellow ball from urn B)
- \( P(B) = \frac{1}{3} \) (Probability of picking urn B)
- \( P(Y) \) (Probability of picking a yellow ball) can be calculated using the law of total probability.
\[ P(Y) = P(Y|A) \times P(A) + P(Y|B) \times P(B) + P(Y|C) \times P(C) \]
\[ P(Y) = \frac{9}{15} \times \frac{1}{3} + \frac{5}{17} \times \frac{1}{3} + \frac{11}{23} \times \frac{1}{3} \]
\[ P(Y) = \frac{3}{5} \times \frac{1}{3} + \frac{5}{17} \times \frac{1}{3} + \frac{11}{23} \times \frac{1}{3} \]
\[ P(Y) = \frac{1}{5} + \frac{5}{51} + \frac{11}{69} \]
\[ P(Y) = \frac{13}{23} \]
Now, substitute these values into Bayes' theorem:
\[ P(B|Y) = \frac{\frac{5}{17} \times \frac{1}{3}}{\frac{13}{23}} \]
\[ P(B|Y) = \frac{5}{17} \times \frac{1}{3} \times \frac{23}{13} \]
\[ P(B|Y) = \frac{5}{51} \times 23 \]
\[ P(B|Y) = \frac{115}{51} \]
So, the probability that the chosen ball came from urn B given that it was a yellow ball is \( \frac{115}{51} \).
