One boy can solve `60%`of the problem in a book and another can solve `80%`. The probability that at least one of the two can solve a problem chosen at random from the book is .

Options

A. `2//25`
B. `23//25`
C. `4//25`
D. `9//10`.

Answer

To solve the problem, we need to find the probability that at least one of the two boys can solve a randomly chosen problem from the book.

Let's denote:
- \( P(A) \): Probability that the first boy can solve a problem = 60% = \( \frac{60}{100} = \frac{3}{5} \)
- \( P(B) \): Probability that the second boy can solve a problem = 80% = \( \frac{80}{100} = \frac{4}{5} \)

We want to find the probability that at least one of the boys can solve the problem. This can be calculated using the complement rule:

\[ P(\text{At least one can solve}) = 1 - P(\text{Neither can solve}) \]

First, we need to find the probabilities that each boy cannot solve the problem:
- \( P(A') \): Probability that the first boy cannot solve a problem = \( 1 - P(A) = 1 - \frac{3}{5} = \frac{2}{5} \)
- \( P(B') \): Probability that the second boy cannot solve a problem = \( 1 - P(B) = 1 - \frac{4}{5} = \frac{1}{5} \)

Next, we calculate the probability that neither boy can solve the problem:
\[ P(A' \cap B') = P(A') \times P(B') = \left(\frac{2}{5}\right) \times \left(\frac{1}{5}\right) = \frac{2}{25} \]

Now, we can find the probability that at least one boy can solve the problem:
\[ P(\text{At least one can solve}) = 1 - P(A' \cap B') = 1 - \frac{2}{25} = \frac{25}{25} - \frac{2}{25} = \frac{23}{25} \]

Thus, the final answer is:
\[ \text{The probability that at least one of the two can solve a problem is } \frac{23}{25}. \]