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Compute the mean and standard deviation of the number of defective chips.

Suppose that 1% of all Olympic athletes use steroids or other performance-enhancing drugs. Because drug

use is a serious violation of Olympic rules, the International Olympic Committee has implemented random

drug tests. Let us assume that the test will report positive with probability 0.9 if the athlete uses drugs,

and with probability 0.15 if the athlete does not use drugs.

a) A randomly selected athlete has a positive drug test. What is the probability that he uses drugs?

b) A randomly selected athlete has a negative drug test. What is the probability that she uses drugs?

c) In fact, an athlete is typically asked to give two samples, A and B, which are tested independently.

Assuming that the results of the A and B tests truly are independent, and that both of an athlete’s samples

are positive, what is the probability that he uses drugs?

d) Given that an athlete’s A test is positive, what is the probability that her B test will also be posi-

tive?

1

For budgeting purposes, each manager in a company must estimate their group’s monthly expenses. Man-

ager 1 estimates that her monthly expenses are uniformly distributed between $5,000 and $45,000. Manager

2 estimates that his monthly expenses are normally distributed with mean $21,000 and standard deviation

$3,000. Manager 3 estimates that her monthly expenses follow a discrete distribution with p($15, 000) = 0.2,

p($20, 000) = 0.4, p($25, 000) = 0.3, and p($30, 000) = 0.1.

a) For each of the three managers, compute the mean and standard deviation of their group’s monthly

expenses.

b) For each of the three managers, estimate the probabilities that their group’s monthly expenses are:

i) between $17,000 and $24,000, ii) higher than $22,000, iii) below $18,000, and iv) exactly $25,000.

c) The company wants to budget enough for each group that the probability of expenses being over budget

is at most 15%. How much should the company budget for each manager?

d) The company wants to budget enough for each group that the probability of expenses being over budget

is at most 1.5%. How much should the company budget for each manager?

A manufacturer produces 1000 computer chips for a mission-critical application. Each chip costs $100

to manufacture and sells for $2000, but has a 1% chance of being defective.

a) Compute the mean and standard deviation of the number of defective chips.

b) Assume that 99.9% of defective chips are discovered (and destroyed) before they are sold, while the

other 0.1% are sold with defects. The manufacturer suffers an estimated loss of $20,000,000 for each defec-

tive chip sold (due to lawsuits, penalties, and bad public relations). What is the manufacturer’s expected

profit?

c) The manufacturer develops a new quality control technology which enables detection of 100% of de-

fective chips. How much would the implementation of this technology increase the manufacturer’s expected

profit?

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