The odds in favour of A solving a problem is 5:7 and odds against B solving the same problem is 9:6 . What is the probability that if both of them try, the problem will be solved?

Option a: 117/180. Solution: Probability that the Problem is Solved Let A and B be the events that student A and student B solve the problem, respectively. Given: - Odds in favor of A solving the problem is 5:7 : P(A) = 5/5+7 = 5/12 ⟹ P(A) = 7/12 - Odds against B solving the problem is 9:6 : P(B) = 9/9+6 = 9/15 = 3/5 ⟹ P(B) = 2/5 The problem will be solved if at least one of them solves it. Using the complement rule: P(solved) = 1 - P(neither solves) Assuming the events are independent: P(neither solves) = P(A) × P(B) = 7/12 × 3/5 = 21/60 Now, compute the probability that the problem is solved: P(solved) = 1 - 21/60 = 39/60 = 117/180 Hence, Option A is the correct answer.

ProbabilityMCQMTP Jun 24 Series IQuestion 3366 of 187

All Questions

The odds in favour of $\displaystyle A$ solving a problem is $\displaystyle 5:7$ and odds against $\displaystyle B$ solving the same problem is $\displaystyle 9:6$ . What is the probability that if both of them try, the problem will be solved?

Options

\displaystyle 117/180

\displaystyle 181/200

\displaystyle 147/180

\displaystyle 119/180

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Correct Answer

✅ Option a — $\displaystyle 117/180$

All Options:

A $\displaystyle 117/180$
B $\displaystyle 181/200$
C $\displaystyle 147/180$
D $\displaystyle 119/180$

Detailed Solution & Explanation

**Probability that the Problem is Solved** Let

\displaystyle A

and

\displaystyle B

be the events that student A and student B solve the problem, respectively. Given: - Odds in favor of A solving the problem is

\displaystyle 5:7

P(A) = \frac{5}{5+7} = \frac{5}{12} \Rightarrow P(\bar{A}) = \frac{7}{12}

- Odds against B solving the problem is

\displaystyle 9:6

P(\bar{B}) = \frac{9}{9+6} = \frac{9}{15} = \frac{3}{5} \Rightarrow P(B) = \frac{2}{5}

The problem will be solved if at least one of them solves it. Using the complement rule:

P(\text{solved}) = 1 - P(\text{neither solves})

Assuming the events are independent:

P(\text{neither solves}) = P(\bar{A}) \times P(\bar{B}) = \frac{7}{12} \times \frac{3}{5} = \frac{21}{60}

Now, compute the probability that the problem is solved:

P(\text{solved}) = 1 - \frac{21}{60} = \frac{39}{60} = \frac{117}{180}

Hence, **Option A** is the correct answer.

About This Chapter: Probability

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Probability Operations, Expected Value

A logic-heavy chapter dealing with random experiments, events (mutually exclusive, exhaustive), set theory probability, conditional probability, and Bayes' Theorem. It forms the basis for Theoretical Distributions.

View Official ICAI Syllabus

Exam Strategy Tip

Always draw a quick Venn Diagram or tree when faced with 'At least one' or 'Only A but not B' wording. It saves you from double-counting.

Key Concepts to Understand

Probability

A numerical measure of the likelihood of an event occurring, expressed as a number between 0 (impossible) and 1 (certain). Formula: P(A) = Number of favorable outcomes / Total number of equally likely outcomes.

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The odds in favour of A\displaystyle AA solving a problem is 5:7\displaystyle 5:75:7 and odds against B\displaystyle BB solving the same problem is 9:6\displaystyle 9:69:6. What is the probability that if both of them try, the problem will be solved?