In a given race, the odds in favourof horses, A,B,C,D are 1:3,1:4,1:5 and 1:6 respectively. Find the probability that one of the them wins the race.

Answer

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

### Step-by-Step Solution:

1. **Understanding Odds**: The odds in favor of a horse represent the ratio of favorable outcomes to unfavorable outcomes. For horse A, the odds are 1:3, which means there is 1 favorable outcome and 3 unfavorable outcomes.

2. **Calculating Total Outcomes for Each Horse**:
- For horse A: Odds = 1:3
- Favorable outcomes = 1
- Unfavorable outcomes = 3
- Total outcomes = 1 + 3 = 4
- For horse B: Odds = 1:4
- Favorable outcomes = 1
- Unfavorable outcomes = 4
- Total outcomes = 1 + 4 = 5
- For horse C: Odds = 1:5
- Favorable outcomes = 1
- Unfavorable outcomes = 5
- Total outcomes = 1 + 5 = 6
- For horse D: Odds = 1:6
- Favorable outcomes = 1
- Unfavorable outcomes = 6
- Total outcomes = 1 + 6 = 7

3. **Calculating Individual Probabilities**:
- Probability of horse A winning:
\[ P(A) = \frac{1}{4} \]
- Probability of horse B winning:
\[ P(B) = \frac{1}{5} \]
- Probability of horse C winning:
\[ P(C) = \frac{1}{6} \]
- Probability of horse D winning:
\[ P(D) = \frac{1}{7} \]

4. **Finding the Combined Probability**:
- The probability that one of the horses A, B, C, or D wins the race is the sum of their individual probabilities:
\[ P(A \cup B \cup C \cup D) = P(A) + P(B) + P(C) + P(D) \]
- Substituting the values:
\[ P(A \cup B \cup C \cup D) = \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} \]

5. **Finding a Common Denominator**:
- The least common multiple (LCM) of 4, 5, 6, and 7 is 420.
- Converting each probability to have a denominator of 420:
- \( \frac{1}{4} = \frac{105}{420} \)
- \( \frac{1}{5} = \frac{84}{420} \)
- \( \frac{1}{6} = \frac{70}{420} \)
- \( \frac{1}{7} = \frac{60}{420} \)

6. **Adding the Probabilities**:
- Now, add the converted fractions:
\[ P(A \cup B \cup C \cup D) = \frac{105 + 84 + 70 + 60}{420} = \frac{319}{420} \]

### Final Answer:
The probability that one of the horses A, B, C, or D wins the race is:
\[ \frac{319}{420} \]