In a given race, the odds in favour of horses`H_1,H_2,H_3"and"H_4` are 1:2,1:3,1:4,1:5 respectively. Find the probability that one of them wins the race.

Options

A. A. `57/60`
B. B. `1/20`
C. C. `2/7`
D. D. `77/60`

Answer

To solve the problem, we need to find the probability that one of the horses \( H_1, H_2, H_3, \) or \( H_4 \) wins the race, given the odds in favor of each horse.

### Step 1: Understand the Odds
The odds in favor of each horse can be expressed as follows:
- For horse \( H_1 \): Odds = 1:2
- For horse \( H_2 \): Odds = 1:3
- For horse \( H_3 \): Odds = 1:4
- For horse \( H_4 \): Odds = 1:5

### Step 2: Convert Odds to Probability
The probability \( P \) of winning for each horse can be calculated using the formula:
\[ P(H) = \frac{\text{Odds in favor}}{\text{Total Odds}} \]
Where Total Odds = Odds in favor + Odds against.

1. For horse \( H_1 \):
\[ P(H_1) = \frac{1}{1 + 2} = \frac{1}{3} \]

2. For horse \( H_2 \):
\[ P(H_2) = \frac{1}{1 + 3} = \frac{1}{4} \]

3. For horse \( H_3 \):
\[ P(H_3) = \frac{1}{1 + 4} = \frac{1}{5} \]

4. For horse \( H_4 \):
\[ P(H_4) = \frac{1}{1 + 5} = \frac{1}{6} \]

### Step 3: Calculate the Total Probability
Now, we need to find the total probability that one of these horses wins the race by summing the individual probabilities:
\[ P(\text{one of them wins}) = P(H_1) + P(H_2) + P(H_3) + P(H_4) \]
Substituting the values we calculated:
\[ P(\text{one of them wins}) = \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} \]

### Step 4: Find a Common Denominator
To add these fractions, we need a common denominator. The least common multiple (LCM) of 3, 4, 5, and 6 is 60.

Now, convert each fraction:
- \( \frac{1}{3} = \frac{20}{60} \)
- \( \frac{1}{4} = \frac{15}{60} \)
- \( \frac{1}{5} = \frac{12}{60} \)
- \( \frac{1}{6} = \frac{10}{60} \)

### Step 5: Add the Converted Fractions
Now we can add them:
\[ P(\text{one of them wins}) = \frac{20}{60} + \frac{15}{60} + \frac{12}{60} + \frac{10}{60} = \frac{57}{60} \]

### Final Step: Conclusion
Thus, the probability that one of the horses \( H_1, H_2, H_3, \) or \( H_4 \) wins the race is:
\[ \frac{57}{60} \]