1. **State the problem:** We have a probability tree diagram involving two people, Jack and Charlie. Jack has a probability of qualifying as $\frac{3}{10}$ and not qualifying as $1 - \frac{3}{10} = \frac{7}{10}$. Charlie's probabilities depend on whether Jack qualifies or not. From the problem, Charlie's probability of not qualifying is $\frac{1}{8}$ in both cases. 2. **Find the probability that both Jack and Charlie qualify:** - Probability Jack qualifies: $\frac{3}{10}$ - Probability Charlie qualifies given Jack qualifies: $1 - \frac{1}{8} = \frac{7}{8}$ Using the multiplication rule for independent branches in a tree: $$P(\text{Jack qualifies and Charlie qualifies}) = \frac{3}{10} \times \frac{7}{8}$$ 3. **Calculate the product:** $$\frac{3}{10} \times \frac{7}{8} = \frac{3 \times 7}{10 \times 8} = \frac{21}{80}$$ 4. **Find the probability that Jack does not qualify and Charlie qualifies:** - Probability Jack does not qualify: $\frac{7}{10}$ - Probability Charlie qualifies given Jack does not qualify: $1 - \frac{1}{8} = \frac{7}{8}$ Multiply: $$\frac{7}{10} \times \frac{7}{8} = \frac{49}{80}$$ 5. **Summary:** - Probability both qualify: $\frac{21}{80}$ - Probability Jack does not qualify and Charlie qualifies: $\frac{49}{80}$ These probabilities can be used to answer further questions about the scenario.