Math Problem Statement

The odds against A solving a problem are 10 to 7 and the odds in favour of B solving the problem are 15 to 12. What is the probability that if both of them try, the problem will be solved?

Solution

The problem asks for the probability that both AA and BB will solve the problem, given certain odds.

Given:

  • The odds against AA solving the problem are 10 to 7.
    • This means the probability of AA solving the problem, P(A)P(A), is: P(A)=710+7=717P(A) = \frac{7}{10 + 7} = \frac{7}{17}
  • The odds in favor of BB solving the problem are 15 to 12.
    • This means the probability of BB solving the problem, P(B)P(B), is: P(B)=1515+12=1527=59P(B) = \frac{15}{15 + 12} = \frac{15}{27} = \frac{5}{9}

To find the probability that both AA and BB solve the problem, we assume their attempts are independent. The probability of both solving it is the product of their individual probabilities: P(both A and B)=P(A)×P(B)=717×59=35153P(\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 AA and BB solve the problem is 35153\frac{35}{153}, which simplifies to 521\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 BB 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 CC solving a problem are 4:5, what is P(C)P(C)?
  4. What is the probability that neither AA nor BB 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.

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Math Problem Analysis

Mathematical Concepts

Probability
Odds

Formulas

P(A) = favorable outcomes / (favorable outcomes + unfavorable outcomes)
P(B) = favorable outcomes / (favorable outcomes + unfavorable outcomes)
P(A and B) = P(A) × P(B)

Theorems

Multiplication rule for independent events

Suitable Grade Level

Grades 9-12