Question

25% of the chips produced by a machine are faulty.
(a) If two chips are chosen at random, what is the probability that neither is faulty?
(b) A sample of 8 chips is taken. Find the probability that exactly 2 chips are faulty.
(c) A sample of 8 chips is taken. Find the probability that no more than 4 chips are faulty.

Answer 1

a) is straighforward enough.
25% are faulty, 75% are not
assuming independence you get \(.75\times .75\) for the first answer

Answer 2

b) is not too bad either
two are faulty, six are not. there there are \(\dbinom{8}{2}=\frac{8\times 7}{2}=28\) ways to arrange the 2 faulty ones and 6 good ones, and the probability of each such arrangement is
\[.25^2\times .75^6\] so your answer is
\[25\times .25^2\times .75^6\]
this is the binomial probability, the probability of \(k\) successes in \(n\) indepenent trials with the probability of success as \(p\) is
\[P(x=k)=\dbinom{n}{k}p^k(1-p)^{n-k}\]

Answer 3

oops typo there, answer is
\[28(.25)^2(.75)^6\]

Answer 4

last one is a drag
you have to compute
\[P(x=0),P(x=1),P(x=2),P(x=3),P(x=4)\] and add