1. **State the problem:** We need to find the probability of event $C$, denoted as $P(C)$, using the given tree diagram. 2. **Understand the tree diagram:** - The first branching splits into events $A$ and $B$, each with probability $\frac{1}{2}$. - From $A$, the branches go to $C$ with probability $\frac{1}{3}$ and to $D$ with probability $\frac{2}{3}$. - From $B$, the branches go to $C$ with probability $\frac{1}{3}$ and to $D$ with probability $\frac{2}{3}$. 3. **Formula for total probability:** $$ P(C) = P(A) \times P(C|A) + P(B) \times P(C|B) $$ This means the total probability of $C$ is the sum of the probabilities of reaching $C$ through $A$ and through $B$. 4. **Substitute the values:** $$ P(C) = \frac{1}{2} \times \frac{1}{3} + \frac{1}{2} \times \frac{1}{3} $$ 5. **Calculate each term:** $$ P(C) = \frac{1}{6} + \frac{1}{6} $$ 6. **Add the fractions:** $$ P(C) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} $$ 7. **Simplify the fraction:** $$ P(C) = \frac{\cancel{2}}{\cancel{6}} = \frac{1}{3} $$ **Final answer:** $$ P(C) = \frac{1}{3} $$