We are given the following information:

• Urn 1 contains 5 white balls and 3 red balls.
• Urn 2 contains 5 white balls and 6 red balls.
• The probability of selecting Urn 1 is $P(U_1) = 0.75$.
• The probability of selecting Urn 2 is $P(U_2) = 1 - 0.75 = 0.25$.
• A ball is drawn, and it is white.

We are asked to find the probability that Urn 1 was selected given that the ball drawn is white. This is a conditional probability problem, and we can use Bayes' Theorem to solve it:

$P(U_1 \mid W) = \frac{P(W \mid U_1) P(U_1)}{P(W)}$

Where:

• $P(U_1 \mid W)$ is the probability that Urn 1 was selected given that the ball drawn is white.
• $P(W \mid U_1)$ is the probability of drawing a white ball from Urn 1.
• $P(U_1)$ is the probability of selecting Urn 1.
• $P(W)$ is the total probability of drawing a white ball from either urn.

Step 1: Calculate $P(W \mid U_1)$

From Urn 1, there are 5 white balls out of 8 total balls. Therefore: $P(W \mid U_1) = \frac{5}{8}$

Step 2: Calculate $P(W \mid U_2)$

From Urn 2, there are 5 white balls out of 11 total balls. Therefore: $P(W \mid U_2) = \frac{5}{11}$

Step 3: Calculate $P(W)$

The total probability of drawing a white ball, $P(W)$, is given by the law of total probability: $P(W) = P(W \mid U_1) P(U_1) + P(W \mid U_2) P(U_2)$ Substituting the values we have: $P(W) = \left( \frac{5}{8} \times 0.75 \right) + \left( \frac{5}{11} \times 0.25 \right)$ $P(W) = 0.46875 + 0.113636 = 0.582386$

Step 4: Apply Bayes' Theorem

Now we can apply Bayes' Theorem: $P(U_1 \mid W) = \frac{P(W \mid U_1) P(U_1)}{P(W)} = \frac{\frac{5}{8} \times 0.75}{0.582386}$ $P(U_1 \mid W) = \frac{0.46875}{0.582386} \approx 0.8047$

Thus, the probability that Urn 1 was selected given that the ball drawn is white is approximately 0.8047.

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Would you like more details or explanations on any part of this process? Here are some related questions for further learning:

1. How does Bayes' Theorem apply to other conditional probability problems?
2. What is the law of total probability and how is it used in this case?
3. How would the solution change if the probabilities of selecting each urn were different?
4. Can Bayes' Theorem be applied to continuous distributions?
5. What is the interpretation of $P(U_1 \mid W)$ in real-world terms?

Tip: Always check the conditions for applying Bayes' Theorem: you need the prior probabilities, likelihoods, and the total probability of the evidence.