Of all the articles in a box, 80% are satisfactory, while 20% are not. The probability of
obtaining exactly five good items out of eight randomly selected articles is

Options

A. `0.003`
B. `0.013`
C. `0.132`
D. `0.147`

Explanation

To solve the problem of finding the probability of obtaining exactly five satisfactory items out of eight randomly selected articles, we can use the binomial probability formula. Here’s a step-by-step solution:

### Step 1: Identify the parameters
We know that:
- The probability of drawing a satisfactory article (success) \( p = 0.8 \)
- The probability of drawing a non-satisfactory article (failure) \( q = 0.2 \)
- The number of trials (articles drawn) \( n = 8 \)
- The number of successes we want (satisfactory articles) \( k = 5 \)

### Step 2: Use the binomial probability formula
The binomial probability formula is given by:
\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]
Where:
- \( \binom{n}{k} \) is the binomial coefficient, which represents the number of ways to choose \( k \) successes in \( n \) trials.

### Step 3: Calculate the binomial coefficient
Calculate \( \binom{8}{5} \):
\[ \binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{8!}{5!3!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]

### Step 4: Calculate \( p^k \) and \( q^{n-k} \)
Now calculate \( p^k \) and \( q^{n-k} \):
\[ p^5 = (0.8)^5 = 0.32768 \]
\[ q^{3} = (0.2)^3 = 0.008 \]

### Step 5: Substitute values into the formula
Now substitute these values into the binomial probability formula:
\[ P(X = 5) = \binom{8}{5} (0.8)^5 (0.2)^3 = 56 \times 0.32768 \times 0.008 \]

### Step 6: Calculate the final probability
Now perform the multiplication:
\[ P(X = 5) = 56 \times 0.32768 \times 0.008 = 56 \times 0.00262144 = 0.1468 \]

### Step 7: Round the answer
Finally, rounding \( 0.1468 \) gives us approximately \( 0.147 \).

Thus, the probability of obtaining exactly five good items out of eight randomly selected articles is:
\[ \boxed{0.147} \]

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