Question

The centers for disease control have determined that when a person is given a vaccine, the probability that the person will develop immunity to a virus is \(0.8\). If \(8\) people are given this vaccine, then the probability that exactly \(4\) will develop immunity is...

• A. \((0.2)(0.8)^4\)
• B. \((0.112)(0.8)^4\)
• C. \((0.112)(0.8)^6\)
• D. \((0.2)^4(0.8)^4\)

Hint:

Use binomial: C(8,4)(0.8)^4(0.2)^4.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Each person independently develops immunity with probability \(p=0.8\), and fails with probability \(q=0.2\). For \(n=8\) people, the number with immunity is binomial.

Step 2: Formula:
\[ P(X=r)=\binom nr p^r q^{n-r} \]

Step 3: Substitute:
\[ P(X=4)=\binom84(0.8)^4(0.2)^4 \]
\(\binom84=70\) and \((0.2)^4=0.0016\), so \(70\times0.0016=0.112\).

Step 4: Result:
\[ P(X=4)=(0.112)(0.8)^4 \]
Option (B).

Final Answer:
The probability is 0.112 x (0.8)^4. \[ \boxed{(0.112)(0.8)^4} \]