If the probabiliy of a horse A winning a race is `(1)/(5)` and the probability of horse B winning the same race is `(1)/(4)` , what is the probability that one of the horses will win ?

Answer

To solve the problem, we need to find the probability that either horse A or horse B wins the race. Let's go through the solution step by step.

### Step 1: Define the Events
Let:
- Event A = Horse A wins the race
- Event B = Horse B wins the race

### Step 2: Write Down the Given Probabilities
From the problem, we know:
- Probability of horse A winning, \( P(A) = \frac{1}{5} \)
- Probability of horse B winning, \( P(B) = \frac{1}{4} \)

### Step 3: Identify the Type of Events
Since horse A and horse B cannot win at the same time (only one horse can win the race), the events A and B are mutually exclusive. This means:
- \( P(A \cap B) = 0 \) (the probability that both A and B win is zero)

### Step 4: Use the Formula for the Probability of the Union of Two Events
To find the probability that either horse A or horse B wins, we use the formula for the probability of the union of two mutually exclusive events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Since \( P(A \cap B) = 0 \), the formula simplifies to:
\[ P(A \cup B) = P(A) + P(B) \]

### Step 5: Substitute the Values
Now we can substitute the values we have:
\[ P(A \cup B) = P(A) + P(B) = \frac{1}{5} + \frac{1}{4} \]

### Step 6: Find a Common Denominator
To add the fractions, we need a common denominator. The least common multiple of 5 and 4 is 20. So we convert the fractions:
\[ P(A) = \frac{1}{5} = \frac{4}{20} \]
\[ P(B) = \frac{1}{4} = \frac{5}{20} \]

### Step 7: Add the Probabilities
Now we can add the two probabilities:
\[ P(A \cup B) = \frac{4}{20} + \frac{5}{20} = \frac{9}{20} \]

### Step 8: Conclusion
The probability that either horse A or horse B will win the race is:
\[ \boxed{\frac{9}{20}} \]