In a series of 3 one-day cricket matches between teams A and B of a college, the probability of team A winning or drawing are 1/3 and 1/6 respectively. If a win, loss or draw gives 2, 0 and I point respectively, then what is the probability that team A will score 5 points in the series?

Options

A. `(17)/(18)`
B. `(11)/(12)`
C. `(1)/(12)`
D. `(1)/(18)`

Explanation

To solve the problem, we need to find the probability that team A scores exactly 5 points in a series of 3 one-day cricket matches against team B.

### Step-by-Step Solution:

1. **Understanding the Points System**:
- Win (W) = 2 points
- Draw (D) = 1 point
- Loss (L) = 0 points

2. **Probabilities Given**:
- Probability of team A winning a match, \( P(W) = \frac{1}{3} \)
- Probability of team A drawing a match, \( P(D) = \frac{1}{6} \)
- Probability of team A losing a match, \( P(L) = 1 - P(W) - P(D) = 1 - \frac{1}{3} - \frac{1}{6} = \frac{1}{2} \)

3. **Finding Cases for Scoring 5 Points**:
To score exactly 5 points, team A can achieve this through the following combinations of wins and draws:
- Case 1: Win, Win, Draw (W, W, D) → Points = 2 + 2 + 1 = 5
- Case 2: Win, Draw, Win (W, D, W) → Points = 2 + 1 + 2 = 5
- Case 3: Draw, Win, Win (D, W, W) → Points = 1 + 2 + 2 = 5

There are no other combinations that can yield exactly 5 points because any loss would contribute 0 points.

4. **Calculating the Probability for Each Case**:
- **Case 1 (W, W, D)**:
\[ P(W, W, D) = P(W) \times P(W) \times P(D) = \frac{1}{3} \times \frac{1}{3} \times \frac{1}{6} = \frac{1}{54} \]

- **Case 2 (W, D, W)**:
\[ P(W, D, W) = P(W) \times P(D) \times P(W) = \frac{1}{3} \times \frac{1}{6} \times \frac{1}{3} = \frac{1}{54} \]

- **Case 3 (D, W, W)**:
\[ P(D, W, W) = P(D) \times P(W) \times P(W) = \frac{1}{6} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{54} \]

5. **Total Probability of Scoring 5 Points**:
Now, we add the probabilities of all three cases:
\[ P(5 \text{ points}) = P(W, W, D) + P(W, D, W) + P(D, W, W) = \frac{1}{54} + \frac{1}{54} + \frac{1}{54} = \frac{3}{54} = \frac{1}{18} \]

### Final Answer:
The probability that team A will score exactly 5 points in the series is \( \frac{1}{18} \).