An institution contracted to buy books from two suppliers P and Q. P supplies 60% and Q supplies 40% of the books. Of P's books, 96% are good and of Q's books, 72% are good. The probability that a randomly chosen book, which is found good, is made by Q is ??

To solve the problem, we need to determine the probability that a randomly chosen book, which is found to be good, was supplied by Q. We'll use Bayes' theorem to solve this problem.

### Given Information:
- Probability that a book is supplied by P, \( P(P) = 0.60 \)
- Probability that a book is supplied by Q, \( P(Q) = 0.40 \)
- Probability that a book supplied by P is good, \( P(G | P) = 0.96 \)
- Probability that a book supplied by Q is good, \( P(G | Q) = 0.72 \)

### Required Probability:
We need to find the probability that a book was supplied by Q given that it is good, i.e., \( P(Q | G) \).

### Applying Bayes' Theorem:
\[
P(Q | G) = \frac{P(G | Q) \cdot P(Q)}{P(G)}
\]

Where \( P(G) \) is the total probability that a book is good, which can be calculated as:
\[
P(G) = P(G | P) \cdot P(P) + P(G | Q) \cdot P(Q)
\]

Substituting the given values:
\[
P(G) = (0.96 \times 0.60) + (0.72 \times 0.40)
\]

Now, let's calculate \( P(G) \):
\[
P(G) = (0.576) + (0.288) = 0.864
\]

### Calculate \( P(Q | G) \):
\[
P(Q | G) = \frac{0.72 \times 0.40}{0.864} = \frac{0.288}{0.864} = \frac{1}{3} \approx 0.333
\]

### Final Answer:
The probability that a randomly chosen book, which is found to be good, was supplied by Q is approximately **0.333** or **33.3%**.

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Do you want more details on this or have any questions? 

Here are five related questions:
1. How would the probability change if Q supplied 50% of the books?
2. What is the probability that a randomly chosen book was supplied by P and is good?
3. How would you approach if Q’s books had a different good percentage?
4. What is the probability that a randomly chosen book is bad?
5. How does Bayes' theorem help in real-world decision-making?

**Tip:** Bayes' theorem is crucial in updating probabilities based on new information, making it highly applicable in fields like diagnostics and machine learning.

Understand how to calculate the probability that a good book was supplied by Supplier Q using Bayes' Theorem. Explore detailed calculations and concepts in probability, suitable for advanced high school students.

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