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: Count arrangements:
Choose which 4 of the 8 people develop immunity: $\binom84=70$ ways.

Step 2: Probability of one arrangement:
Four succeed and four fail: $(0.8)^4(0.2)^4$.

Step 3: Multiply:
$70\times(0.2)^4=70\times0.0016=0.112$. So the probability is $0.112\times(0.8)^4$, option (B).

Final Answer:
70 arrangements times (0.8)^4 (0.2)^4. \[ \boxed{B} \]