If the probability of a horse A winning a race is 1/4 and the probability of a horse B winning the same race is 1/5, then the probability that either of them will win the race is

Answer

To find the probability that either horse A or horse B will win the race, we can use the formula for the probability of the union of two mutually exclusive events. Here are the steps to solve the problem:

### Step 1: Identify the probabilities
We are given:
- Probability of horse A winning, \( P(A) = \frac{1}{4} \)
- Probability of horse B winning, \( P(B) = \frac{1}{5} \)

### Step 2: Determine if the events are mutually exclusive
Since horse A and horse B cannot win at the same time, these events are mutually exclusive. Therefore, the probability of both A and B winning at the same time, \( P(A \cap B) \), is 0.

### Step 3: Use the formula for the union of two events
The probability that either horse A or horse B wins is given by:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the values we have:
\[ P(A \cup B) = P(A) + P(B) - 0 \]
\[ P(A \cup B) = \frac{1}{4} + \frac{1}{5} \]

### Step 4: Find a common denominator
To add the fractions, we need a common denominator. The least common multiple of 4 and 5 is 20.
- Convert \( \frac{1}{4} \) to a fraction with a denominator of 20:
\[ \frac{1}{4} = \frac{5}{20} \]
- Convert \( \frac{1}{5} \) to a fraction with a denominator of 20:
\[ \frac{1}{5} = \frac{4}{20} \]

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

### Step 6: Convert to decimal (if needed)
To express \( \frac{9}{20} \) as a decimal:
\[ \frac{9}{20} = 0.45 \]

### Final Answer
The probability that either horse A or horse B will win the race is \( \frac{9}{20} \) or 0.45.

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