Question

The probability that a person who undergoes a certain operation will survive is 0.2. If 5 patients undergo similar operations, find the probability that exactly four will survive.

• A. \( 0.0042 \)
• B. \( 0.0084 \)
• C. \( 0.0032 \)
• D. \( 0.0064 \)

Hint:

In binomial probability problems, always identify \( n \), \( p \), and \( r \) correctly before substituting into the formula.

Answer 1

The Correct Option is D

Step 1: Identify the probability model.
This is a binomial distribution problem since there are a fixed number of trials and two outcomes (survive or not survive).

Step 2: Assign values.
Probability of survival \( p = 0.2 \), Probability of not surviving \( q = 1 - 0.2 = 0.8 \), Number of trials \( n = 5 \), Required survivors \( r = 4 \).

Step 3: Apply the binomial probability formula.
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \] \[ P(X = 4) = \binom{5}{4} (0.2)^4 (0.8)^1 \] \[ = 5 \times 0.0016 \times 0.8 = 0.0064 \]

Step 4: Final conclusion.
The probability that exactly four patients survive is \[ \boxed{0.0064} \]