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?