1. **Problem Statement:** Calculate the marginal probabilities from the joint probability table for events $A_1, A_2, A_3$ and $B_1, B_2$. 2. **Given Joint Probabilities:** $$\begin{array}{c|ccc} & A_1 & A_2 & A_3 \\ \hline B_1 & 0.1 & 0.3 & 0.2 \\ B_2 & 0.2 & 0.1 & 0.1 \end{array}$$ 3. **Marginal Probability Formula:** The marginal probability of an event is the sum of the joint probabilities over the other variable. - For $P(A_i)$: sum over all $B_j$, $$P(A_i) = \sum_j P(A_i \cap B_j)$$ - For $P(B_j)$: sum over all $A_i$, $$P(B_j) = \sum_i P(A_i \cap B_j)$$ 4. **Calculate Marginal Probabilities for $A_i$:** - $P(A_1) = P(A_1 \cap B_1) + P(A_1 \cap B_2) = 0.1 + 0.2 = 0.3$ - $P(A_2) = 0.3 + 0.1 = 0.4$ - $P(A_3) = 0.2 + 0.1 = 0.3$ 5. **Calculate Marginal Probabilities for $B_j$:** - $P(B_1) = 0.1 + 0.3 + 0.2 = 0.6$ - $P(B_2) = 0.2 + 0.1 + 0.1 = 0.4$ 6. **Calculate Conditional Probabilities:** - $P(A_1|B_1) = \frac{P(A_1 \cap B_1)}{P(B_1)} = \frac{0.1}{0.6} = \frac{1}{6} \approx 0.1667$ - $P(A_2|B_1) = \frac{0.3}{0.6} = \frac{1}{2} = 0.5$ 7. **Note:** $P(B_3|A_1)$ is requested but $B_3$ is not defined in the table, so it cannot be calculated. **Final answers:** - Marginal probabilities: $P(A_1)=0.3$, $P(A_2)=0.4$, $P(A_3)=0.3$, $P(B_1)=0.6$, $P(B_2)=0.4$ - Conditional probabilities: $P(A_1|B_1)=0.1667$, $P(A_2|B_1)=0.5$, $P(B_3|A_1)$ undefined.