It is estimated that 1.3% of the matches produced at a factory are damaged in some way. A household box contains 462 matches.

a. Calculate the expected number of damaged matches in a household box.

b. Find the variance of the number of damaged matches and the variance of the number of undamaged matches in a household box.

c. Show that approximately 10.4% of the household boxes are expected to contain exactly eight damaged matches.

d. Calculate the probability that at least one from a sample of two household boxes contains exactly eight damaged matches.