In a race the probabilities of A and B winning the race are `1/3 and 1/6` respectively. Find the probability of neighter of them winning the race.

Answer

To find the probability of neither A nor B winning the race, we can follow these steps:

### Step 1: Identify the probabilities of A and B winning
We are given:
- Probability of A winning, \( P(A) = \frac{1}{3} \)
- Probability of B winning, \( P(B) = \frac{1}{6} \)

### Step 2: Calculate the probability of either A or B winning
To find the probability of either A or B winning, we can use the formula for the union of two mutually exclusive events:
\[ P(A \cup B) = P(A) + P(B) \]
Substituting the values we have:
\[ P(A \cup B) = \frac{1}{3} + \frac{1}{6} \]

### Step 3: Find a common denominator
To add the fractions, we need a common denominator. The least common multiple of 3 and 6 is 6.
- Convert \( \frac{1}{3} \) to sixths:
\[ \frac{1}{3} = \frac{2}{6} \]
- Now we can add:
\[ P(A \cup B) = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \]

### Step 4: Calculate the probability of neither A nor B winning
The probability of neither A nor B winning can be found using the complement rule:
\[ P(\text{neither A nor B}) = 1 - P(A \cup B) \]
Substituting the value we found:
\[ P(\text{neither A nor B}) = 1 - \frac{1}{2} = \frac{1}{2} \]

### Final Answer
The probability of neither A nor B winning the race is:
\[ \frac{1}{2} \]

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