1. **State the problem:** We want to find the probability that Rosie does her homework given the probabilities related to her going to the beach and doing homework. 2. **Given probabilities:** - Probability Rosie goes to the beach: $P(B) = \frac{6}{11}$ - Probability Rosie does homework if she goes to the beach: $P(H|B) = \frac{1}{2}$ - Probability Rosie does homework if she does not go to the beach: $P(H|B^c) = \frac{3}{4}$ - Probability Rosie does not go to the beach: $P(B^c) = 1 - P(B) = 1 - \frac{6}{11} = \frac{5}{11}$ 3. **Formula used:** The total probability that Rosie does her homework is given by the law of total probability: $$ P(H) = P(H|B)P(B) + P(H|B^c)P(B^c) $$ 4. **Calculate each term:** $$ P(H|B)P(B) = \frac{1}{2} \times \frac{6}{11} = \frac{6}{22} $$ $$ P(H|B^c)P(B^c) = \frac{3}{4} \times \frac{5}{11} = \frac{15}{44} $$ 5. **Add the probabilities:** $$ P(H) = \frac{6}{22} + \frac{15}{44} $$ To add, find a common denominator: $$ \frac{6}{22} = \frac{12}{44} $$ So, $$ P(H) = \frac{12}{44} + \frac{15}{44} = \frac{27}{44} $$ 6. **Final answer:** The probability that Rosie does her homework is $$ \boxed{\frac{27}{44}} $$