The problem asks for the probability that both $A$ and $B$ will solve the problem, given certain odds.

Given:

• The odds against $A$ solving the problem are 10 to 7.
  • This means the probability of $A$ solving the problem, $P(A)$, is: $P(A) = \frac{7}{10 + 7} = \frac{7}{17}$
• The odds in favor of $B$ solving the problem are 15 to 12.
  • This means the probability of $B$ solving the problem, $P(B)$, is: $P(B) = \frac{15}{15 + 12} = \frac{15}{27} = \frac{5}{9}$

To find the probability that both $A$ and $B$ solve the problem, we assume their attempts are independent. The probability of both solving it is the product of their individual probabilities: $P(\text{both A and B}) = P(A) \times P(B) = \frac{7}{17} \times \frac{5}{9} = \frac{35}{153}$

Thus, the probability that both $A$ and $B$ solve the problem is $\frac{35}{153}$, which simplifies to $\frac{5}{21}$.

Would you like further explanation or help with anything else?

Here are 5 related questions for further practice:

1. What are the odds against $B$ solving the problem?
2. How do the odds of solving a problem relate to the probability of solving it?
3. If the odds in favor of $C$ solving a problem are 4:5, what is $P(C)$?
4. What is the probability that neither $A$ nor $B$ solves the problem?
5. How would dependent events affect the calculation of probabilities?

Tip: When working with odds, remember that the total number of possible outcomes is the sum of the odds in favor and against.