How do I solve this question?
A school lab has sixteen computers. A teacher observes that, in the long run, in 80% of school days, at least 1 machine is not working properly. Assuming the probability of a computer not working properly is independent of the others, find the probability that:
a) a randomly chosen computer is not working in any school day
b) at least 2 computers are not working properly in any school day
Attempt: I’m really not sure how to approach this question. If for 80% of school days at least 1 computer doesnt work that means for 20% of school says at least 15 doesnt work. Now do you use the binomial expansion to find for all school days?
We know that $0.8 = P(\text{broken PCs } \geq 1)$. So:
$$0.2 = P(\text{broken PCs } < 1) = P(\text{broken PCs }= 0) = (1-p)^{16},$$ where $p$ is the probability for the $j$-th computer to be broken. This $p$ is the answer for the first question.
For the second one: we want to calculate $q =P(\text{broken PCs } \geq 2)$. So:
$$1-q = P(\text{broken PCs } < 2) = P(\text{broken PCs }= 0) + P(\text{broken PCs }= 1) = (1-p)^{16} + \binom{16}{1}p(1-p)^{15}.$$
This $q$ is the answer to your second question.