Question

The probability that a certain kind of component will survive a given shock test is \[\frac{3}{4} .\]  Find the probability that among 5 components tested exactly 2 will survive .

 

Answer 1

Let X denote the number of components that survive shock.
Then, X follows  a binomial distribution with n = 5.
Let p be the probability that a certain kind of component will survive a given shock test. 

\[\therefore p = \frac{3}{4}\text{ and } q = \frac{1}{4}\]
\[\text{ Hence, the disrtibution is given by } \]
\[P(X = r) = ^{5}{}{C}_r \left( \frac{3}{4} \right)^r \left( \frac{1}{4} \right)^{5 - r} , r = 0, 1, 2, 3, 4, 5\]
\[ P(\text{ exactly 2 will survive} ) = P(X = 2) \]
\[ = ^{5}{}{C}_2 \left( \frac{3}{4} \right)^2 \left( \frac{1}{4} \right)^{5 - 2} \]
\[ = \frac{10 \times 9}{1024}\]
\[ = 0 . 0879\]