1. a. The probability distribution is more or less given already. There are 5 possible outcomes of the experiment which tests the computers for defects.
: 0 computers are defective
: 1 computer is defective
: 2 computers are defective
: 3 computers are defective
: 4 computers are defective
You define a probability distribution such that . Your probability space is the set of all events of some experiment, so it might be more precise: computer 1 has the defect, computer 2 has the defect, computer 3 has the defect, computer 4 has the defect. Then the events above are some subsets (elements of the power set of the 4 outcomes).
Since you are given probabilities of the events your probability distribution is simply .
b. Given a probability distribution you can find expectation. It is given by
c. I suspect this is asking you to find the standard deviation of the distribution. It is given by (or the variance which is just this number squared)
d. Think about this as the probability of finding exactly 0 defective computers plus the probability of finding exactly 1 defective computer plus the probability of finding exactly 2 defective computers.
2. a. Assuming that the probability of each random person having been in an accident is independent of the probabilities of any others being in an accident this should be easy. Likewise for part b, just phrase the question as the union of disjoint events.