Please, can someone help me with the following problems:

1) A biotechnology manufacturing firm can produce diagnostic test kits at a cost of

$20. Each kit for which there is a demand in the week of production can be sold

for $100. However, the half-life of components in the kit requires the kit to be

scrapped if it is not sold in the week of production. The cost of scrapping the kit

is $5. The weekly demand is summarized as follows:

Weekly demand

Number of units 0 50 100 200

Probability of demand 0.05 0.4 0.3 0.25

How many kits should be produced each week to maximize the mean earnings of the firm?

2) The probability that a lab specimen contains high levels of contamination is 0.10. Five samples are checked, and the samples are independent.

a. What is the probability that none contains high levels of contamination?

b. What is the probability that exactly one contains high levels of

contamination?

c. What is the probability that at least one contains high levels of

contamination?

3) A steel plate contains 20 bolts. Assume that 5 bolts are not torqued to the proper limit. Four bolts are selected at random, without replacement, to be checked for torque.

a. What is the probability that all four of the selected bolts are torqued to the

proper limit?

b. What is the probability that at least one of the selected bolts is not torqued

to the proper limit?