A can solve 80% of the problems given in a book and B can solve 60%. What is the probability that, at least one of them will solve a problem selected at random from the book?

Options

A. `(12)/(25)`
B. `(97)/(100)`
C. `(23)/(25)`
D. `(11)/(25)`

Answer

To find the probability that at least one of A or B will solve a problem selected at random from the book, we can follow these steps:

### Step 1: Define the probabilities
- Let \( P(E) \) be the probability that A solves a problem. Given that A can solve 80% of the problems, we have:
\[ P(E) = 0.80 \]
- Let \( P(F) \) be the probability that B solves a problem. Given that B can solve 60% of the problems, we have:
\[ P(F) = 0.60 \]

### Step 2: Calculate the probabilities that A and B do not solve the problem
- The probability that A does not solve the problem is:
\[ P(E') = 1 - P(E) = 1 - 0.80 = 0.20 \]
- The probability that B does not solve the problem is:
\[ P(F') = 1 - P(F) = 1 - 0.60 = 0.40 \]

### Step 3: Calculate the probability that neither A nor B solves the problem
Since A and B are independent, the probability that neither A nor B solves the problem is:
\[ P(E' \cap F') = P(E') \times P(F') = 0.20 \times 0.40 = 0.08 \]

### Step 4: Calculate the probability that at least one of them solves the problem
The probability that at least one of them solves the problem is given by:
\[ P(E \cup F) = 1 - P(E' \cap F') = 1 - 0.08 = 0.92 \]

### Step 5: Convert the probability to a fraction
To express 0.92 as a fraction:
\[ 0.92 = \frac{92}{100} = \frac{23}{25} \]

### Final Answer
Thus, the probability that at least one of A or B will solve a problem selected at random from the book is:
\[ \frac{23}{25} \]