A die is thrown 10 times. If getting an even number is considered as a success, then the probability of four successes is

Options

A. `.^(10)C_(4)(1/2)^(4)`
B. `.^(10)C_(4)(1/2)^(6)`
C. `.^(10)C_(4)(1/2)^(8)`
D. `.^(10)C_(6)(1/2)^(10)`

Answer

To solve the problem step by step, we will use the concept of Bernoulli trials.

### Step 1: Identify the parameters
We are throwing a die 10 times.
- Total trials (n) = 10
- Success (getting an even number) = 4
- Therefore, r (number of successes) = 4

### Step 2: Determine the probability of success and failure
When rolling a die, the even numbers are 2, 4, and 6. Thus:
- Probability of success (getting an even number) = P(success) = 3/6 = 1/2
- Probability of failure (getting an odd number) = P(failure) = 3/6 = 1/2

### Step 3: Use the binomial probability formula
The formula for the probability of getting exactly r successes in n trials is given by:
\[ P(X = r) = \binom{n}{r} p^r (1-p)^{n-r} \]
Where:
- \( \binom{n}{r} \) is the binomial coefficient (n choose r)
- p is the probability of success
- (1-p) is the probability of failure

Substituting the values:
- n = 10
- r = 4
- p = 1/2
- (1-p) = 1/2

### Step 4: Calculate the binomial coefficient
\[ \binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!} \]

### Step 5: Calculate the probabilities
Now substituting into the formula:
\[ P(X = 4) = \binom{10}{4} \left(\frac{1}{2}\right)^4 \left(\frac{1}{2}\right)^{10-4} \]
\[ P(X = 4) = \binom{10}{4} \left(\frac{1}{2}\right)^{10} \]

### Step 6: Simplify the expression
Since \( \left(\frac{1}{2}\right)^{10} = \frac{1}{1024} \):
\[ P(X = 4) = \binom{10}{4} \cdot \frac{1}{1024} \]

### Step 7: Calculate \( \binom{10}{4} \)
Calculating \( \binom{10}{4} \):
\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]

### Step 8: Final probability calculation
Now substituting back:
\[ P(X = 4) = 210 \cdot \frac{1}{1024} = \frac{210}{1024} \]

### Step 9: Simplify the fraction
The fraction can be simplified:
\[ \frac{210}{1024} = \frac{105}{512} \]

Thus, the probability of getting exactly 4 successes (even numbers) when a die is thrown 10 times is:
\[ \frac{105}{512} \]