Discrete Binomial question probability that one component does not work

Question

The probability that a component passes quality assurance tests is 0.92. Twelve components are selected at random. What is the probability that at least eleven out of the twelve pass quality assurance tests?

Given the question above, I attempted it like this: I found out the probability of one components not working, and subtraction from all components which are working.

P(1 component not working) =$\binom{12}{1}\cdot0.08^1\cdot(1-0.08)^{11} = 0.384$

So the probability of 1 working is $1-0.38 = 0.62$

Is this correct?

Answer 1

You're almost there! You've made a small mistake though, which even @antkam forgot to mention. With your method, using the complement won't yield the right answer, since, if we let X be the number of components which failed QA tests;

• $X\sim B(12, 0.08)$
• P(At least 11 components passed QA tests) = P(At most 1 component failed QA tests)

So by that logic, the probability you're looking for is $P(x\leq1)=P(x=0)+P(x=1)$, and NOT $1-P(x=1)$.

Merry Christmas! :)