1. **Problem statement:** We have two factories, A and B, producing red and black widgets with given probabilities. We select one factory uniformly at random, then sample two widgets from that factory. Both widgets are red. We want to find the probability that these two red widgets came from Factory A. 2. **Given data:** - Factory A: $P(\text{red})=0.4$, $P(\text{black})=0.6$ - Factory B: $P(\text{red})=0.8$, $P(\text{black})=0.2$ - Probability of choosing Factory A or B: $P(A)=P(B)=0.5$ 3. **Goal:** Find $P(A \mid \text{both red})$. 4. **Formula:** Use Bayes' theorem: $$ P(A \mid \text{both red}) = \frac{P(\text{both red} \mid A) P(A)}{P(\text{both red})} $$ where $$ P(\text{both red}) = P(\text{both red} \mid A) P(A) + P(\text{both red} \mid B) P(B) $$ 5. **Calculate $P(\text{both red} \mid A)$:** Since widgets are sampled independently, $$ P(\text{both red} \mid A) = P(\text{red} \mid A)^2 = 0.4^2 = 0.16 $$ 6. **Calculate $P(\text{both red} \mid B)$:** $$ P(\text{both red} \mid B) = 0.8^2 = 0.64 $$ 7. **Calculate $P(\text{both red})$:** $$ P(\text{both red}) = 0.16 \times 0.5 + 0.64 \times 0.5 = 0.08 + 0.32 = 0.4 $$ 8. **Apply Bayes' theorem:** $$ P(A \mid \text{both red}) = \frac{0.16 \times 0.5}{0.4} = \frac{0.08}{0.4} = 0.2 $$ 9. **Express as a fraction:** $$ 0.2 = \frac{1}{5} $$ **Final answer:** The probability that the two red widgets came from Factory A is $\frac{1}{5}$. Numerator: 1 Denominator: 5