1. **State the problem:** We have a test with two sections, A and B. - 85% pass section A. - Of those who pass section A, 78% pass section B. - Of those who fail section A, 36% fail section B. We need to complete the tree diagram with all probabilities. 2. **Identify given probabilities:** - $P(\text{Pass A}) = 0.85$ - $P(\text{Fail A}) = 1 - 0.85 = 0.15$ - $P(\text{Pass B} | \text{Pass A}) = 0.78$ - $P(\text{Fail B} | \text{Fail A}) = 0.36$ 3. **Calculate missing conditional probabilities:** - $P(\text{Fail B} | \text{Pass A}) = 1 - 0.78 = 0.22$ - $P(\text{Pass B} | \text{Fail A}) = 1 - 0.36 = 0.64$ 4. **Calculate joint probabilities for each branch:** - $P(\text{Pass A and Pass B}) = P(\text{Pass A}) \times P(\text{Pass B} | \text{Pass A}) = 0.85 \times 0.78 = 0.663$ - $P(\text{Pass A and Fail B}) = 0.85 \times 0.22 = 0.187$ - $P(\text{Fail A and Pass B}) = 0.15 \times 0.64 = 0.096$ - $P(\text{Fail A and Fail B}) = 0.15 \times 0.36 = 0.054$ 5. **Summary of completed tree diagram probabilities:** - Section A: Pass = 0.85, Fail = 0.15 - Section B given Pass A: Pass = 0.78, Fail = 0.22 - Section B given Fail A: Pass = 0.64, Fail = 0.36 All probabilities sum to 1: $0.663 + 0.187 + 0.096 + 0.054 = 1$ This completes the tree diagram with all probabilities filled in.