India plays two matches each with West Indies and Australia. In any match the probabilities of India getting points 0,1 and 2 are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, the probability of India getting at least 7 points is

Options

A. `0*8750`
B. `*0875`
C. `0*0625`
D. `0*0250`

Answer

To solve the problem, we need to calculate the probability of India getting at least 7 points from 4 matches against West Indies and Australia, where the probabilities of getting 0, 1, or 2 points in each match are given as follows:

- Probability of 0 points (P(0)) = 0.45
- Probability of 1 point (P(1)) = 0.05
- Probability of 2 points (P(2)) = 0.50

### Step 1: Understanding the total points
India plays a total of 4 matches, and the maximum points that can be scored in these matches is 8 (if India scores 2 points in all matches). We need to find the probability of India scoring at least 7 points, which means we need to consider the scenarios where India scores exactly 7 points and exactly 8 points.

### Step 2: Calculate the probability of scoring exactly 7 points
To score exactly 7 points, India can achieve this in two ways:
1. Scoring 2 points in 3 matches and 1 point in 1 match.
2. Scoring 2 points in all 4 matches (which gives 8 points).

#### Case 1: 2 points in 3 matches and 1 point in 1 match
- The number of ways to choose 3 matches out of 4 to score 2 points is given by the binomial coefficient \( \binom{4}{3} \).
- The probability of scoring 2 points in 3 matches and 1 point in 1 match is:

\[ P(2, 2, 2, 1) = \binom{4}{3} \cdot (0.50)^3 \cdot (0.05)^1 \]

Calculating this:
- \( \binom{4}{3} = 4 \)
- \( (0.50)^3 = 0.125 \)
- \( (0.05)^1 = 0.05 \)

So,
\[ P(2, 2, 2, 1) = 4 \cdot 0.125 \cdot 0.05 = 0.025 \]

#### Case 2: 2 points in all 4 matches
- The probability of scoring 2 points in all matches is:

\[ P(2, 2, 2, 2) = \binom{4}{4} \cdot (0.50)^4 = 1 \cdot (0.50)^4 \]

Calculating this:
- \( (0.50)^4 = 0.0625 \)

### Step 3: Total probability of scoring at least 7 points
Now, we can sum the probabilities from both cases:
\[ P(\text{at least 7 points}) = P(7) + P(8) = 0.025 + 0.0625 = 0.0875 \]

### Final Answer
The probability of India getting at least 7 points is \( \boxed{0.0875} \).