In a horse race the odds in favour of three horses are 1 : 2, 1 : 3 and 1 : 4. The probability that one of the horse will win the race is

Options

A. `(37)/(60)`
B. `(47)/(60)`
C. `(1)/(4)`
D. `(3)/(4)`

Answer

To solve the problem, we first need to determine the probabilities of each horse winning based on the given odds.

### Step 1: Understand the Odds
The odds in favor of the horses A, B, and C are given as:
- Horse A: 1 : 2
- Horse B: 1 : 3
- Horse C: 1 : 4

### Step 2: Convert Odds to Probability
The odds of a horse winning can be converted to probability using the formula:
\[ \text{Probability} = \frac{\text{Odds in favor}}{\text{Total Odds}} \]

For each horse:
1. **Horse A**: The odds are 1 : 2, which means for every 1 win, there are 2 losses. Therefore, the total outcomes = 1 (win) + 2 (losses) = 3.
\[ P(A) = \frac{1}{1 + 2} = \frac{1}{3} \]

2. **Horse B**: The odds are 1 : 3, which means for every 1 win, there are 3 losses. Therefore, the total outcomes = 1 (win) + 3 (losses) = 4.
\[ P(B) = \frac{1}{1 + 3} = \frac{1}{4} \]

3. **Horse C**: The odds are 1 : 4, which means for every 1 win, there are 4 losses. Therefore, the total outcomes = 1 (win) + 4 (losses) = 5.
\[ P(C) = \frac{1}{1 + 4} = \frac{1}{5} \]

### Step 3: Calculate the Total Probability
Since the events are mutually exclusive (only one horse can win), we can simply add the probabilities of each horse winning:
\[ P(A \cup B \cup C) = P(A) + P(B) + P(C) \]
Substituting the values we found:
\[ P(A \cup B \cup C) = \frac{1}{3} + \frac{1}{4} + \frac{1}{5} \]

### Step 4: Find a Common Denominator
To add these fractions, we need a common denominator. The least common multiple of 3, 4, and 5 is 60.
- Convert each fraction:
\[ P(A) = \frac{1}{3} = \frac{20}{60} \]
\[ P(B) = \frac{1}{4} = \frac{15}{60} \]
\[ P(C) = \frac{1}{5} = \frac{12}{60} \]

### Step 5: Add the Probabilities
Now, we can add the converted probabilities:
\[ P(A \cup B \cup C) = \frac{20}{60} + \frac{15}{60} + \frac{12}{60} = \frac{47}{60} \]

### Final Answer
The probability that one of the horses will win the race is:
\[ \boxed{\frac{47}{60}} \]