Tutorial Problems
Problem 1. Suppose an insurance company classifies people into one of the three classes: good
risks, average risks and bad risks. Their records indicate that the probabilities that good, avg and
bad risk persons will be involved in an accident over a 1-year period span are respectively 0.05,
0.15 and 0.30. If 20 percent of the population are good risks, 50 percent are average risks and 30
percent are bad risks, what proportion of people have accidents in a fixed year? If policy holder A
had no accidents in 1997, what is the probability that he or she was a good risk. Similarly compute
the probability that he or she was an average risk given that Ahad no accidents.
Problem 2. A parallel system functions whenever at least one of its component works. Consider
a parallel system of ncomponents and suppose that each component independently works with
probability 1
2. Find the conditional probability that component 1 works given that the system is
functioning.
Problem 3. Independent trials that result in a success with probability pand a failure with
probability 1 −pare called Bernoulli trials. Let Pndenote the probability that nBernoulli trials
result in an even number of successses (0 being considered an even number). Show that for all
n≥1,
Pn=p(1 −Pn−1) + (1 −p)Pn−1(1)
and use this to prove (by induction) that Pn=1 + (1 −2p)n
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Problem 4. Prove directly that P(E|F) = P(E|F G)P(G|F) + P(E|F Gc)P(Gc|F)
Problem 5. Urn Acontains 2 white balls and 1 black ball, whereas urn Bcontains 1 white ball
and 5 black balls. A ball is drawn at random from urn Aand placed in urn B. A ball is then drawn
from urn B. It happens to be white. What is the probability that the ball transferred was white?
Problem 6 Let Aand Bbe events having positive probability. State whether each of the following
statements in (i) necessarily true, (ii) necessarily false, or (iii) possibly true.
1. If Aand Bare mutually exclusive, then they are independent.
2. If Aand Bare independent, then they are mutually exclusive.
3. P(A) = P(B)=0.6, and Aand Bare mutually exclusive.
4. P(A) = P(B)=0.6 and Aand Bare independent.
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