The odds against P solving a problem are 8 : 6 and odds in favour of Q solving the same problem are 14 : 10 The probability of the problem being solved, if both of them try it, is

Options

A. `5/21`
B. `16/21`
C. `5/12`
D. `5/7`

Answer

To solve the problem, we need to determine the probability of the problem being solved by either P or Q. We will follow these steps:

### Step 1: Determine the probability of P solving the problem
The odds against P solving the problem are given as 8:6. This means:
- For every 8 failures, there are 6 successes.

To convert odds into probability, we can use the formula:
\[ P(P \text{ solves}) = \frac{\text{Successes}}{\text{Successes} + \text{Failures}} = \frac{6}{8 + 6} = \frac{6}{14} = \frac{3}{7} \]

### Step 2: Determine the probability of Q solving the problem
The odds in favor of Q solving the problem are given as 14:10. This means:
- For every 14 successes, there are 10 failures.

Using the same formula for probability:
\[ P(Q \text{ solves}) = \frac{\text{Successes}}{\text{Successes} + \text{Failures}} = \frac{14}{14 + 10} = \frac{14}{24} = \frac{7}{12} \]

### Step 3: Determine the probability of neither P nor Q solving the problem
To find the probability of the problem being solved by either P or Q, we first need to find the probability of neither solving it:
\[ P(\text{neither P nor Q solves}) = (1 - P(P \text{ solves})) \times (1 - P(Q \text{ solves})) \]
Calculating each term:
\[ 1 - P(P \text{ solves}) = 1 - \frac{3}{7} = \frac{4}{7} \]
\[ 1 - P(Q \text{ solves}) = 1 - \frac{7}{12} = \frac{5}{12} \]
Now, multiply these probabilities:
\[ P(\text{neither P nor Q solves}) = \frac{4}{7} \times \frac{5}{12} = \frac{20}{84} = \frac{5}{21} \]

### Step 4: Calculate the probability of the problem being solved
Now, we can find the probability of the problem being solved by either P or Q:
\[ P(\text{either P or Q solves}) = 1 - P(\text{neither P nor Q solves}) = 1 - \frac{5}{21} = \frac{21 - 5}{21} = \frac{16}{21} \]

### Final Answer
The probability of the problem being solved if both P and Q try it is:
\[ \frac{16}{21} \]