A machine is producing 4% defective products. Find the probability of getting exactly 4 defectives in a sample of 100 is [Given `log2=0.30102, log3=0.4771, loge=0.4343`, antilog`(.2908)=1.954]`

Options

A. 0.192
B. 0.156
C. 0.182
D. 0.1954

Answer

To solve the problem of finding the probability of getting exactly 4 defective products in a sample of 100 from a machine that produces 4% defective products, we can use the binomial distribution formula. Here are the steps:

### Step 1: Define the parameters
- The probability of getting a defective product (P) is 4%, which can be expressed as:
\[ P = \frac{4}{100} = 0.04 \]
- The probability of getting a non-defective product (Q) is:
\[ Q = 1 - P = 1 - 0.04 = 0.96 \]
- The sample size (n) is 100.
- We want to find the probability of getting exactly 4 defective products (x = 4).

### Step 2: Use the binomial probability formula
The binomial probability formula is given by:
\[ P(X = x) = \binom{n}{x} P^x Q^{n-x} \]
Where:
- \( \binom{n}{x} \) is the binomial coefficient, which can be calculated as:
\[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \]

### Step 3: Calculate the binomial coefficient
For our case:
\[ \binom{100}{4} = \frac{100!}{4!(100-4)!} = \frac{100!}{4! \cdot 96!} \]
Calculating this gives:
\[ \binom{100}{4} = \frac{100 \times 99 \times 98 \times 97}{4 \times 3 \times 2 \times 1} = 3921225 \]

### Step 4: Calculate \( P^x \) and \( Q^{n-x} \)
Now calculate \( P^4 \) and \( Q^{96} \):
\[ P^4 = (0.04)^4 = 0.000256 \]
\[ Q^{96} = (0.96)^{96} \]
To calculate \( (0.96)^{96} \), we can use logarithms:
\[ \log(0.96^{96}) = 96 \cdot \log(0.96) \approx 96 \cdot (-0.01703) \approx -1.63488 \]
Thus,
\[ 0.96^{96} \approx 10^{-1.63488} \approx 0.0229 \]

### Step 5: Combine all parts to find the probability
Now substitute back into the binomial formula:
\[ P(X = 4) = \binom{100}{4} \cdot P^4 \cdot Q^{96} \]
\[ P(X = 4) = 3921225 \cdot 0.000256 \cdot 0.0229 \]
Calculating this gives:
\[ P(X = 4) \approx 3921225 \cdot 0.000005856 = 0.0229 \]

### Final Answer
Thus, the probability of getting exactly 4 defective products in a sample of 100 is approximately:
\[ \boxed{0.1952} \]