In a given race the odds in favour of three horses A, B, C are 1 : 3, 1 : 4, 1 : 5 respectively. Assuming that dead head is impossible the probability that one of them wins is

Options

A. `(7)/(60)`
B. `(37)/(60)`
C. `(1)/(5)`
D. `(1)/(8)`

Answer

To solve the problem, we need to find the probability that one of the horses A, B, or C wins the race, given the odds in favor of each horse.

### Step-by-Step Solution:

1. **Understanding Odds**:
- The odds in favor of a horse represent the ratio of the probability of the horse winning to the probability of it losing.
- For horse A, the odds are 1:3, meaning for every 1 time A wins, it loses 3 times. Thus, the total outcomes for A are 1 (win) + 3 (loss) = 4.
- Therefore, the probability of A winning (P(A)) is given by:
\[ P(A) = \frac{1}{1 + 3} = \frac{1}{4} \]

2. **Calculating Probability for Horse B**:
- The odds in favor of horse B are 1:4. So, the total outcomes for B are 1 (win) + 4 (loss) = 5.
- Therefore, the probability of B winning (P(B)) is:
\[ P(B) = \frac{1}{1 + 4} = \frac{1}{5} \]

3. **Calculating Probability for Horse C**:
- The odds in favor of horse C are 1:5. Thus, the total outcomes for C are 1 (win) + 5 (loss) = 6.
- Therefore, the probability of C winning (P(C)) is:
\[ P(C) = \frac{1}{1 + 5} = \frac{1}{6} \]

4. **Finding the Total Probability**:
- Since we are assuming that one of the horses must win (no dead heat), we can add the probabilities of each horse winning:
\[ P(\text{one of A, B, or C wins}) = P(A) + P(B) + P(C) \]
- Substituting the values we found:
\[ P(\text{one of A, B, or C wins}) = \frac{1}{4} + \frac{1}{5} + \frac{1}{6} \]

5. **Finding a Common Denominator**:
- The least common multiple (LCM) of 4, 5, and 6 is 60.
- Now we convert each fraction:
\[ \frac{1}{4} = \frac{15}{60}, \quad \frac{1}{5} = \frac{12}{60}, \quad \frac{1}{6} = \frac{10}{60} \]

6. **Adding the Probabilities**:
- Now we can add the converted fractions:
\[ P(\text{one of A, B, or C wins}) = \frac{15}{60} + \frac{12}{60} + \frac{10}{60} = \frac{37}{60} \]

### Final Answer:
The probability that one of the horses A, B, or C wins the race is:
\[ \frac{37}{60} \]