Course: Mathematics-I (CDA101), Topic- Probability
Tutorial Sheet #1
1. We are given a number of darts. When we throw a dart at a target, we have a
probability of 1/4 of hitting the target. What is the probability of obtaining at
least one hit if three darts are thrown? Calculate this probability two ways. Are all
outcomes in the sample space equally likely?
2. Two balls are selected sequentially (without replacement) from an urn containing
three red, four white, and five blue balls. (a) What is the probability that the first
is red and the second is blue? (b) What is the probability of selecting a white ball
on the second draw if the first ball is replaced before the second is selected? (c)
What is the probability of selecting a white ball on the second draw if the first ball
is not replaced before the second is selected?
3. Manufacturer X produces personal computers (PCs) at two different locations in the
world. Fifteen percent of the PCs produced at location A are delivered defective
to a retail outlet, while 5 percent of the PCs produced at location B are delivered
defective to the same retail store. If the manufacturing plant at A produces 1,000,000
PCs per year and the plant at B produces 150,000 PCs per year, find the probability
of purchasing a defective PC.
4. The events A and B are mutually exclusive. Can they be independent? Explain
with reasoning.
5. How many equations do you need to establish the independence of n events?
6. If A ∩ B = {φ}, then what is the relation between the probabilities of A and B c .
7. An urn contains b black balls and r red balls. One of the balls is drawn at random,
but when it is put back in the urn c additional balls of the same colour are put in
with it. Now suppose that we draw another ball. What is the probability that the
first ball drawn was black given that the second ball drawn was red?
8. Three prisoners are informed by their jailer that one of them has been chosen at
random to be executed, and the other two are to be freed. Prisoner A asks the
jailer to tell him privately which of his fellow prisoners will be set free, claiming
that there would be no harm in divulging this information, since he already knows
that at least one will go free. The jailer refuses to answer this question, pointing
out that if A knew which of his fellows were to be set free, then his own probability
of being executed would rise from 1/3 to 1/2, since he would then be one of two
prisoners. What do you think of the jailers reasoning?
9. (a) A gambler has in his pocket a fair coin and a two-headed coin. He selects one of
the coins at random, and when he flips it, it shows heads. What is the probability
that it is the fair coin? (b) Suppose that he flips the same coin a second time and
again it shows heads. Now what is the probability that it is the fair coin? (c)
Suppose that he flips the same coin a third time and it shows tails. Now what is
the probability that it is the fair coin?
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Tutorial Sheet #1
1. We are given a number of darts. When we throw a dart at a target, we have a
probability of 1/4 of hitting the target. What is the probability of obtaining at
least one hit if three darts are thrown? Calculate this probability two ways. Are all
outcomes in the sample space equally likely?
2. Two balls are selected sequentially (without replacement) from an urn containing
three red, four white, and five blue balls. (a) What is the probability that the first
is red and the second is blue? (b) What is the probability of selecting a white ball
on the second draw if the first ball is replaced before the second is selected? (c)
What is the probability of selecting a white ball on the second draw if the first ball
is not replaced before the second is selected?
3. Manufacturer X produces personal computers (PCs) at two different locations in the
world. Fifteen percent of the PCs produced at location A are delivered defective
to a retail outlet, while 5 percent of the PCs produced at location B are delivered
defective to the same retail store. If the manufacturing plant at A produces 1,000,000
PCs per year and the plant at B produces 150,000 PCs per year, find the probability
of purchasing a defective PC.
4. The events A and B are mutually exclusive. Can they be independent? Explain
with reasoning.
5. How many equations do you need to establish the independence of n events?
6. If A ∩ B = {φ}, then what is the relation between the probabilities of A and B c .
7. An urn contains b black balls and r red balls. One of the balls is drawn at random,
but when it is put back in the urn c additional balls of the same colour are put in
with it. Now suppose that we draw another ball. What is the probability that the
first ball drawn was black given that the second ball drawn was red?
8. Three prisoners are informed by their jailer that one of them has been chosen at
random to be executed, and the other two are to be freed. Prisoner A asks the
jailer to tell him privately which of his fellow prisoners will be set free, claiming
that there would be no harm in divulging this information, since he already knows
that at least one will go free. The jailer refuses to answer this question, pointing
out that if A knew which of his fellows were to be set free, then his own probability
of being executed would rise from 1/3 to 1/2, since he would then be one of two
prisoners. What do you think of the jailers reasoning?
9. (a) A gambler has in his pocket a fair coin and a two-headed coin. He selects one of
the coins at random, and when he flips it, it shows heads. What is the probability
that it is the fair coin? (b) Suppose that he flips the same coin a second time and
again it shows heads. Now what is the probability that it is the fair coin? (c)
Suppose that he flips the same coin a third time and it shows tails. Now what is
the probability that it is the fair coin?
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