question_answer

The odds against A solving a certain problem are. 4 to 3 and the odds in favour of B solving the same problem are 7 to 5. Find the probability that the problem will be solved.

Answer:

We are given that the odds against A are 4 to 3. \[\therefore \]      \[P(A)=\frac{3}{4+3}=\frac{3}{7}\] It is also given that the odds in favour of B are 7 to 5. \[\therefore \]      \[P(B)=\frac{7}{7+5}=\frac{7}{12}\] The problem will be solved if at least one of them solves the problem. So, we have to find \[P(A\cup B).\]    Since, A and B are independent events.   \[\therefore P(A\cup B)=1-P(\bar{A})P(\bar{B})=1-\left( 1-\frac{3}{7} \right)\left( 1-\frac{7}{12} \right)=\frac{16}{21}\]Hence, the required probability is 16/21.