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 at most 3 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.

\[\therefore p = \frac{3}{4}\text{ and } q = \frac{1}{4}\]
\[\text{ Hence, the disrtibution is given by } \]
\[P(X = r) = ^{5}{}{C}_r \left( \frac{3}{4} \right)^r \left( \frac{1}{4} \right)^{5 - r} , r = 0, 1, 2, 3, 4, 5\]

\[ P(\text{ atmost 3 will survive} ) = P(X \leq 3)\]
\[ = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)\]
\[ = ^{5}{}{C}_{0} \left( \frac{3}{4} \right)^0 \left( \frac{1}{4} \right)^{5 - 0} + ^{5}{}{C}_1 \left( \frac{3}{4} \right)^1 \left( \frac{1}{4} \right)^{5 - 1} + ^{5}{}{C}_2 \left( \frac{3}{4} \right)^2 \left( \frac{1}{4} \right)^{5 - 2} + ^{5}{}{C}_3 \left( \frac{3}{4} \right)^3 \left( \frac{1}{4} \right)^{5 - 3} \]
\[ = \left( \frac{1}{4} \right)^5 + 5\left( \frac{3}{4} \right) \left( \frac{1}{4} \right)^4 + 10 \left( \frac{3}{4} \right)^2 \left( \frac{1}{4} \right)^3 + 10 \left( \frac{3}{4} \right)^3 \left( \frac{1}{4} \right)^2 \]
\[ = \frac{1 + 15 + 90 + 270}{1024}\]
\[ = \frac{376}{1024}\]
\[ = 0 . 3672\]