If chance of A , winning a certain race be `(1)/(6)` and the chance of B winning it is `(1)/(3)` , what is the chance that neither should win ?

Answer

To solve the problem step by step, we will first identify the probabilities of A and B winning, then calculate the probabilities of A and B losing, and finally determine the probability that neither A nor B wins.

### Step 1: Identify the probabilities of winning
- The probability of A winning the race, denoted as P(A), is given as \( P(A) = \frac{1}{6} \).
- The probability of B winning the race, denoted as P(B), is given as \( P(B) = \frac{1}{3} \).

**Hint:** Remember that the probability of an event is a number between 0 and 1, where 0 means the event will not happen and 1 means it will definitely happen.

### Step 2: Calculate the probabilities of losing
- The probability of A losing the race, denoted as P(A loses), is given by:
\[ P(A \text{ loses}) = 1 - P(A) = 1 - \frac{1}{6} = \frac{5}{6} \]

- The probability of B losing the race, denoted as P(B loses), is given by:
\[ P(B \text{ loses}) = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3} \]

**Hint:** The probability of losing is simply 1 minus the probability of winning.

### Step 3: Calculate the probability that neither A nor B wins
- The probability that neither A nor B wins can be calculated as:
\[ P(\text{neither A nor B wins}) = P(A \text{ loses}) \times P(B \text{ loses}) \]

Substituting the values we found:
\[ P(\text{neither A nor B wins}) = \left(\frac{5}{6}\right) \times \left(\frac{2}{3}\right) \]

### Step 4: Perform the multiplication
- Now we multiply the fractions:
\[ P(\text{neither A nor B wins}) = \frac{5 \times 2}{6 \times 3} = \frac{10}{18} \]

### Step 5: Simplify the fraction
- We can simplify \( \frac{10}{18} \) by dividing both the numerator and denominator by their greatest common divisor, which is 2:
\[ P(\text{neither A nor B wins}) = \frac{10 \div 2}{18 \div 2} = \frac{5}{9} \]

### Final Answer
The probability that neither A nor B wins the race is \( \frac{5}{9} \).

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