Question

The probability that a certain kind of component will survive a given shock test is \(\frac{3}{4}\). The probability that exactly 2 of the next 4 components tested survive is

• A. \(\frac{9}{41}\)
• B. \(\frac{25}{128}\)
• C. \(\frac{1}{5}\)
• D. \(\frac{27}{128}\)

Hint:

For binomial probability: \(P(X=r)=\binom{n}{r}p^rq^{n-r}\).

Answer 1

The Correct Option is D

Step 1: Identify distribution.
Each component either survives or fails.
So binomial distribution applies.
Step 2: Define parameters.
\[ n=4,\quad p=\frac{3}{4},\quad q=1-p=\frac{1}{4} \]
Step 3: Probability of exactly 2 successes.
\[ P(X=2)=\binom{4}{2}p^2q^2 \]
Step 4: Substitute values.
\[ P(X=2)=6\left(\frac{3}{4}\right)^2\left(\frac{1}{4}\right)^2 \]
\[ =6\cdot \frac{9}{16}\cdot \frac{1}{16} =\frac{54}{256} =\frac{27}{128} \]
Final Answer:
\[ \boxed{\frac{27}{128}} \]