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

Answer

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

### Step-by-Step Solution:

1. **Understanding Odds**: The odds in favor of an event are given in the form of \( a : b \), which means the probability of the event occurring is \( \frac{a}{a+b} \).

2. **Calculating Probabilities**:
- For Horse A: The odds are \( 1 : 3 \).
- Probability \( P(A) = \frac{1}{1+3} = \frac{1}{4} \).
- For Horse B: The odds are \( 1 : 4 \).
- Probability \( P(B) = \frac{1}{1+4} = \frac{1}{5} \).
- For Horse C: The odds are \( 1 : 5 \).
- Probability \( P(C) = \frac{1}{1+5} = \frac{1}{6} \).
- For Horse D: The odds are \( 1 : 6 \).
- Probability \( P(D) = \frac{1}{1+6} = \frac{1}{7} \).

3. **Finding Total Probability**: To find the probability that at least one horse wins, we can sum the probabilities of each horse winning:
\[ P(A \cup B \cup C \cup D) = P(A) + P(B) + P(C) + P(D) \]
Substituting the values we calculated:
\[ P(A \cup B \cup C \cup D) = \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} \]

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

5. **Adding the Fractions**:
\[ P(A \cup B \cup C \cup D) = \frac{105 + 84 + 70 + 60}{420} = \frac{319}{420} \]

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