In a series of 3 one-day cricket matches between teams A and B of a college, the probability of team A winning or drawings are `(1)/(3)` and `(1)/(6)` respectively. If A win, lose or draw gives 2, 0 and 1 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)`

Answer

To find the probability that team A will score 5 points in the series of 3 matches, we need to analyze the scoring system and the probabilities given.

### Step-by-Step Solution:

1. **Understand the Points System**:
- Team A earns:
- 2 points for a win (W)
- 0 points for a loss (L)
- 1 point for a draw (D)

2. **Determine Combinations for 5 Points**:
- To score exactly 5 points, team A can achieve this by winning 2 matches and drawing 1 match. The possible combinations of outcomes (W, D) in 3 matches are:
- WW(D)
- W(D)W
- DWW

3. **Calculate the Probability of Each Combination**:
- The probabilities given are:
- Probability of winning (P(W)) = 1/3
- Probability of drawing (P(D)) = 1/6
- Probability of losing (P(L)) = 1 - P(W) - P(D) = 1 - 1/3 - 1/6 = 1/2

4. **Calculate the Probability for Each Outcome**:
- For each combination, we will calculate the probability:
- For WW(D):
\[ P(WWD) = P(W) \times P(W) \times P(D) = \left(\frac{1}{3}\right) \times \left(\frac{1}{3}\right) \times \left(\frac{1}{6}\right) = \frac{1}{54} \]
- For W(D)W:
\[ P(WDW) = P(W) \times P(D) \times P(W) = \left(\frac{1}{3}\right) \times \left(\frac{1}{6}\right) \times \left(\frac{1}{3}\right) = \frac{1}{54} \]
- For DWW:
\[ P(DWW) = P(D) \times P(W) \times P(W) = \left(\frac{1}{6}\right) \times \left(\frac{1}{3}\right) \times \left(\frac{1}{3}\right) = \frac{1}{54} \]

5. **Sum the Probabilities**:
- Now, we sum the probabilities of all combinations that yield 5 points:
\[ P(5 \text{ points}) = P(WWD) + P(WDW) + P(DWW) = \frac{1}{54} + \frac{1}{54} + \frac{1}{54} = \frac{3}{54} = \frac{1}{18} \]

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

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