1. Let's explore a random problem involving a tree diagram. 2. Problem: Suppose you have a bag with 3 red balls and 2 blue balls. You draw two balls one after the other without replacement. What is the probability of drawing a red ball first and a blue ball second? 3. To solve this, we use the multiplication rule for probabilities in a tree diagram: $$P(A \text{ and } B) = P(A) \times P(B|A)$$ where $P(B|A)$ is the probability of event B given event A has occurred. 4. Step 1: Probability of drawing a red ball first: $$P(\text{Red}_1) = \frac{3}{5}$$ because there are 3 red balls out of 5 total. 5. Step 2: After drawing one red ball, there are now 4 balls left (2 red, 2 blue). Probability of drawing a blue ball second: $$P(\text{Blue}_2|\text{Red}_1) = \frac{2}{4} = \frac{1}{2}$$ 6. Step 3: Multiply these probabilities to find the combined probability: $$P(\text{Red}_1 \text{ and } \text{Blue}_2) = \frac{3}{5} \times \frac{1}{2} = \frac{3}{10}$$ 7. Therefore, the probability of drawing a red ball first and a blue ball second without replacement is $$\boxed{\frac{3}{10}}$$. This problem illustrates how tree diagrams help visualize sequential events and calculate combined probabilities step-by-step.