MAS3301 Bayesian Statistics
Problems 1 and Solutions
Semester 2
2008-9
Problems 1
1. Let E1 , E2 , E3 be events. Let I1 , I2 , I3 be the corresponding indicators so that I1 = 1 if E1
occurs and I1 = 0 otherwise.
(a) Let IA = 1 − (1 − I1 )(1 − I2 ). Verify that IA is the indicat or for the event A where
A = (E1 ∨ E2 ) (that is “E1 or E2 ”) and show that
Pr(A) = Pr(E1 ) + Pr(E2 ) − Pr(E1 ∧ E2 )
where (E1 ∧ E2 ) is “E1 and E2 ”.
(b) Find a formula, in terms of I1 , I2 , I3 for IB , the indicator for the event B where B =
(E1 ∨E2 ∨E3 ) and derive a formula for Pr(B) in terms of Pr(E1 ), Pr(E2 ), Pr(E3 ), Pr(E1 ∧
E2 ), Pr(E1 ∧ E3 ), Pr(E2 ∧ E3 ), Pr(E1 ∧ E2 ∧ E3 ).
2. In a certain place it rains on one third of the days. The local evening newspaper attempts to
predict whether or not it will rain the following day. Three quarters of rainy days and three
fifths of dry days are correctly predicted by the previous evening’s paper. Given that this
evening’s paper predicts rain, what is the probability that it will actually rain tomorrow?
3. A machine is built to make mass-produced items. Each item made by the machine has a
probability p of being defective. Given the value of p, the items are independent of each
other. Because of the way in which the machines are made, p could take one of several
values. In fact p = X/100 where X has a discrete uniform distribution on the interval [0, 5].
The machine is tested by counting the number of items made before a defective is produced.
Find the conditional probability distribution of X given that the first defective item is the
thirteenth to be made.
4. There are five machines in a factory. Of these machines, three are working
properly and two are defective. Machines which are working properly produce articles each of
which has independently a probability of 0.1 of being imperfect. For the defective machines
this probability is 0.2.
A machine is chosen at random and five articles produced by the machine are examined.
What is the probability that the machine chosen is defective given that, of the five articles
examined, two are imperfect and three are perfect?
5. A crime has been committed. Assume that the crime was committed by exactly one person,
that there are 1000 people who could have committed the crime and that, in the absence of
any evidence, these people are all equally likely to be guilty of the crime.
A piece of evidence is found. It is judged that this evidence would have a probability of 0.99
of being observed if the crime were committed by a particular individual, A, but a probability
of only 0.0001 of being observed if the crime were committed by any other individual.
Find the probability, given the evidence, that A committed the crime.
1
, 6. In an experiment on extra-sensory perception (ESP) a person, A, sits in a sealed room and
points at one of four cards, each of which shows a different picture. In another sealed room
a second person, B, attempts to select, from an identical set of four cards, the card at which
A is pointing. This experiment is repeated ten times and the correct card is selected four
times.
Suppose that we consider three possible states of nature, as follows.
State 1 : There is no ESP and, whichever card A chooses, B is equally likely to select any
one of the four cards. That is, subject B has a probability of 0.25 of selecting the correct
card.
Before the experiment we give this state a probability of 0.7.
State 2 : Subject B has a probability of 0.50 of selecting the correct card.
Before the experiment we give this state a probability of 0.2.
State 3 : Subject B has a probability of 0.75 of selecting the correct card.
Before the experiment we give this state a probability of 0.1.
Assume that, given the true state of nature, the ten trials can be considered to be indepen-
dent.
Find our probabilities after the experiment for the three possible states of nature.
Can you think of a reason, apart from ESP, why the probability of selecting the correct card
might be greater than 0.25?
7. In a certain small town there are n taxis which are clearly numbered 1, 2, . . . , n. Before we
visit the town we do not know the value of n but our probabilities for the possible values of
n are as follows.
n 0 1 2 3 4
Probability 0.00 0.11 0.12 0.13 0.14
n 5 6 7 8 ≥9
Probability 0.14 0.13 0.12 0.11 0.00
On a visit to the town we take a taxi which we assume would be equally likely to be any of
taxis 1, 2, . . . , n. It is taxi number 5. Find our new probabilities for the value of n.
8. A dishonest gambler has a box containing 10 dice which all look the same. However there
are actually three types of dice.
• There are 6 dice of type A which are fair dice with Pr(6 | A) = 1/6 (where Pr(6 | A) is
the probability of getting a 6 in a throw of a type A die).
• There are 2 dice of type B which are biassed with Pr(6 | B) = 0.8.
• There are 2 dice of type C which are biassed with Pr(6 | C) = 0.04.
The gambler takes a die from the box at random and rolls it. Find the conditional probability
that it is of type B given that it gives a 6.
9. In a forest area of Northern Europe there may be wild lynx. At a particular time the number
X of lynx can be between 0 and 5 with
5
Pr(X = x) = 0.6x 0.45−x (x = 0, . . . , 5).
x
A survey is made but the lynx is difficult to spot and, given that the number present is x,
the number Y observed has a probability distribution with
2
Problems 1 and Solutions
Semester 2
2008-9
Problems 1
1. Let E1 , E2 , E3 be events. Let I1 , I2 , I3 be the corresponding indicators so that I1 = 1 if E1
occurs and I1 = 0 otherwise.
(a) Let IA = 1 − (1 − I1 )(1 − I2 ). Verify that IA is the indicat or for the event A where
A = (E1 ∨ E2 ) (that is “E1 or E2 ”) and show that
Pr(A) = Pr(E1 ) + Pr(E2 ) − Pr(E1 ∧ E2 )
where (E1 ∧ E2 ) is “E1 and E2 ”.
(b) Find a formula, in terms of I1 , I2 , I3 for IB , the indicator for the event B where B =
(E1 ∨E2 ∨E3 ) and derive a formula for Pr(B) in terms of Pr(E1 ), Pr(E2 ), Pr(E3 ), Pr(E1 ∧
E2 ), Pr(E1 ∧ E3 ), Pr(E2 ∧ E3 ), Pr(E1 ∧ E2 ∧ E3 ).
2. In a certain place it rains on one third of the days. The local evening newspaper attempts to
predict whether or not it will rain the following day. Three quarters of rainy days and three
fifths of dry days are correctly predicted by the previous evening’s paper. Given that this
evening’s paper predicts rain, what is the probability that it will actually rain tomorrow?
3. A machine is built to make mass-produced items. Each item made by the machine has a
probability p of being defective. Given the value of p, the items are independent of each
other. Because of the way in which the machines are made, p could take one of several
values. In fact p = X/100 where X has a discrete uniform distribution on the interval [0, 5].
The machine is tested by counting the number of items made before a defective is produced.
Find the conditional probability distribution of X given that the first defective item is the
thirteenth to be made.
4. There are five machines in a factory. Of these machines, three are working
properly and two are defective. Machines which are working properly produce articles each of
which has independently a probability of 0.1 of being imperfect. For the defective machines
this probability is 0.2.
A machine is chosen at random and five articles produced by the machine are examined.
What is the probability that the machine chosen is defective given that, of the five articles
examined, two are imperfect and three are perfect?
5. A crime has been committed. Assume that the crime was committed by exactly one person,
that there are 1000 people who could have committed the crime and that, in the absence of
any evidence, these people are all equally likely to be guilty of the crime.
A piece of evidence is found. It is judged that this evidence would have a probability of 0.99
of being observed if the crime were committed by a particular individual, A, but a probability
of only 0.0001 of being observed if the crime were committed by any other individual.
Find the probability, given the evidence, that A committed the crime.
1
, 6. In an experiment on extra-sensory perception (ESP) a person, A, sits in a sealed room and
points at one of four cards, each of which shows a different picture. In another sealed room
a second person, B, attempts to select, from an identical set of four cards, the card at which
A is pointing. This experiment is repeated ten times and the correct card is selected four
times.
Suppose that we consider three possible states of nature, as follows.
State 1 : There is no ESP and, whichever card A chooses, B is equally likely to select any
one of the four cards. That is, subject B has a probability of 0.25 of selecting the correct
card.
Before the experiment we give this state a probability of 0.7.
State 2 : Subject B has a probability of 0.50 of selecting the correct card.
Before the experiment we give this state a probability of 0.2.
State 3 : Subject B has a probability of 0.75 of selecting the correct card.
Before the experiment we give this state a probability of 0.1.
Assume that, given the true state of nature, the ten trials can be considered to be indepen-
dent.
Find our probabilities after the experiment for the three possible states of nature.
Can you think of a reason, apart from ESP, why the probability of selecting the correct card
might be greater than 0.25?
7. In a certain small town there are n taxis which are clearly numbered 1, 2, . . . , n. Before we
visit the town we do not know the value of n but our probabilities for the possible values of
n are as follows.
n 0 1 2 3 4
Probability 0.00 0.11 0.12 0.13 0.14
n 5 6 7 8 ≥9
Probability 0.14 0.13 0.12 0.11 0.00
On a visit to the town we take a taxi which we assume would be equally likely to be any of
taxis 1, 2, . . . , n. It is taxi number 5. Find our new probabilities for the value of n.
8. A dishonest gambler has a box containing 10 dice which all look the same. However there
are actually three types of dice.
• There are 6 dice of type A which are fair dice with Pr(6 | A) = 1/6 (where Pr(6 | A) is
the probability of getting a 6 in a throw of a type A die).
• There are 2 dice of type B which are biassed with Pr(6 | B) = 0.8.
• There are 2 dice of type C which are biassed with Pr(6 | C) = 0.04.
The gambler takes a die from the box at random and rolls it. Find the conditional probability
that it is of type B given that it gives a 6.
9. In a forest area of Northern Europe there may be wild lynx. At a particular time the number
X of lynx can be between 0 and 5 with
5
Pr(X = x) = 0.6x 0.45−x (x = 0, . . . , 5).
x
A survey is made but the lynx is difficult to spot and, given that the number present is x,
the number Y observed has a probability distribution with
2
