Probability of solving of sum correctly by A ,B and C is `(1)/(2),(1)/(3)` and `(1)/(5)` respectively. The probability that at least one of them solves it correctly is

Options

A. `(11)/(15)`
B. `(4)/(15)`
C. `(1)/(20)`
D. `(19)/(20)`

Answer

To find the probability that at least one of A, B, or C solves the problem correctly, we can use the complementary probability approach. This means we will first calculate the probability that none of them solves the problem correctly and then subtract that from 1.

### Step-by-Step Solution:

1. **Identify the probabilities of each person solving the problem correctly:**
- Probability that A solves it correctly, \( P(A) = \frac{1}{2} \)
- Probability that B solves it correctly, \( P(B) = \frac{1}{3} \)
- Probability that C solves it correctly, \( P(C) = \frac{1}{5} \)

2. **Calculate the probabilities of each person not solving the problem correctly:**
- Probability that A does not solve it correctly, \( P(A') = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2} \)
- Probability that B does not solve it correctly, \( P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3} \)
- Probability that C does not solve it correctly, \( P(C') = 1 - P(C) = 1 - \frac{1}{5} = \frac{4}{5} \)

3. **Calculate the probability that none of them solves the problem correctly:**
- The probability that none of them solves it correctly is given by the product of their individual probabilities of not solving it:
\[ P(A' \cap B' \cap C') = P(A') \times P(B') \times P(C') = \frac{1}{2} \times \frac{2}{3} \times \frac{4}{5} \]

4. **Perform the multiplication:**
- First, multiply \( \frac{1}{2} \) and \( \frac{2}{3} \):
\[ \frac{1}{2} \times \frac{2}{3} = \frac{1 \times 2}{2 \times 3} = \frac{2}{6} = \frac{1}{3} \]
- Now multiply \( \frac{1}{3} \) by \( \frac{4}{5} \):
\[ \frac{1}{3} \times \frac{4}{5} = \frac{1 \times 4}{3 \times 5} = \frac{4}{15} \]

5. **Calculate the probability that at least one of them solves the problem correctly:**
- Now, we subtract the probability that none of them solves it from 1:
\[ P(\text{at least one solves}) = 1 - P(A' \cap B' \cap C') = 1 - \frac{4}{15} \]

6. **Perform the subtraction:**
- Convert 1 to a fraction with a denominator of 15:
\[ 1 = \frac{15}{15} \]
- Now subtract:
\[ \frac{15}{15} - \frac{4}{15} = \frac{15 - 4}{15} = \frac{11}{15} \]

### Final Answer:
The probability that at least one of A, B, or C solves the problem correctly is \( \frac{11}{15} \).