Exercises 519
(a) Suppose you decide to stand with your original choice. What are your chances of
winning the car?
(b) Suppose you decide to switch to the remaining door. What are your chances of
winning the car?
(c) Suppose you decide to flip a fair coin. If it comes up heads, you change your
choice; otherwise, you stand pat. What are your chances of winning the car?
- Suppose our manufacturing company purchases a certain part from three different sup-
pliers S1, S 2 , and S 3 .Supplier S1 provides 40% of our parts, and suppliers S2 and S 3
provide 35% and 25%, respectively. Furthermore, 20% of the parts shipped by S are
defective, 10% of the parts shipped by S 2 are defective, and 5% of the parts from S3
are defective. Now, suppose an employee at our company chooses a part at random.
(a) What is the probability that the part is good?
(b) If the part is good, what is the probability that it was shipped by SI?
(c) If the part is defective, what is the probability that it was shipped by Si?
- (a) Give an example that shows three pairwise independent events need not be an
independent set of events.
(b) Give an example that shows three events can be independent without having the
corresponding pairs of events be independent.
- Two manufacturing companies M 1 and M 2 produce a certain unit that is used in an
assembly plant. Company M 1 is larger than M 2 , and it supplies the plant with twice as
many units per day as M 2 does. MI also produces more defects than M 2 .Because of
past experience with these suppliers, it is felt that 10% of MI's units have some defect,
whereas only 5% of M 2 's units are defective. Now, suppose that a unit is selected at
random from a bin in the assembly plant.
(a) What is the probability that the unit was supplied by company M 1?
(b) What is the probability that the unit is defective?
(c) What is the probability that the unit was supplied by M 1 if the unit is defective?
- When a roulette wheel is spun once, there are 38 possible outcomes: 18 red, 18 black,
and 2 green (if the outcome is green the house wins all bets). If a wheel is spun twice,
all 38 • 38 outcomes are equally likely. If you are told that in two spins at least one
resulted in a green outcome, what is the probability that both outcomes were green?
- A computer salesperson makes either one or two sales contacts each day between 1
and 2 PM. If only one contact is made, the probability is 0.2 that a sale will result and
0.8 that no sale will result. If two contacts are made, the two customers make their
decisions independently of each other, each purchasing with probability 0.2 and not
purchasing with probability 0.8. What is the probability that the salesperson has made
two sales this hour?
- Only 1 in 1000 adults is afflicted with a particular rare disease for which a diagnostic
test has been developed. The test is such that when an individual actually has the
disease, a positive result will occur 99% of the time, and an individual without the
disease will show a positive test result only 2% of the time. If a randomly selected
individual is tested and the result is positive, what is the probability that the individual
has the disease? Draw a tree diagram for the problem.
- Suppose three fair coins are tossed. What is the probability that precisely two coins
land heads up if the first coin lands heads up and the second coin lands tails up?
