1. **Stating the problem:** We want to find two probabilities related to significant earnings growth and interest rates changes in 2025: (a) The overall probability of significant earnings growth. (b) Given significant earnings growth, the probability that interest rates will be more than 1% lower than this year. 2. **Define events:** - Let $H$ = interest rates more than 1% higher than this year in 2025. - Let $L$ = interest rates more than 1% lower than this year in 2025. - Let $M$ = interest rates within 1% of this year's rates. - Let $G$ = significant earnings growth. 3. **Given probabilities:** - $P(H) = 0.03$ - $P(L) = 0.85$ - $P(M) = 1 - P(H) - P(L) = 1 - 0.03 - 0.85 = 0.12$ - $P(G|H) = 0.1$ - $P(G|L) = 0.8$ - $P(G|M) = 0.5$ --- 4. **(a) Calculate** $P(G)$, the total probability of significant earnings growth, using the Law of Total Probability: $$ P(G) = P(G|H)P(H) + P(G|L)P(L) + P(G|M)P(M) $$ $$ = (0.1)(0.03) + (0.8)(0.85) + (0.5)(0.12) $$ $$ = 0.003 + 0.68 + 0.06 = 0.743 $$ 5. **(b) Calculate** $P(L|G)$, the probability that interest rates are more than 1% lower given significant growth. Use Bayes' theorem: $$ P(L|G) = \frac{P(G|L)P(L)}{P(G)} $$ $$ = \frac{(0.8)(0.85)}{0.743} = \frac{0.68}{0.743} \approx 0.915 $$ --- **Final answers:** - (a) The probability of significant earnings growth is **$0.743$**. - (b) Given significant growth, the probability interest rates are more than 1% lower is approximately **$0.915$**.