Chapter 5
Conditional probability
In this chapter, we look at more complicated notions of probability, and extend the multiplica
tion rule for probability to cater for events that are
not
independent.
5.1
Introduction
So far we have only considered probabilities of single events or of several independent events,
like two rolls of a die. However, in reality, many events are related. For example, the probability
of it raining in 5 minutes time is dependent on whether or not it is raining now.
We need a mathematical notation to capture how the probability of one event depends on other
events taking place. We do this as follows. Consider two events
A
and
B
. We write
P
(
A

B
)
for the probability of
A
given that
B
has already happened. We describe
P
(
A

B
)
as the
con
ditional probability
of
A
given
B
. For example, the probability of it raining in 5 minutes time
given that it is raining now would be
P
(
Rain in 5 minutes

Raining now
)
.
Example
Utility companies need to be able to forecast periods of high demand. They describe their fore
casts in terms of probabilities. Gas and electricity suppliers might relate them to air temperature.
For example,
P
(
High demand

air temperature is below normal
) = 0
.
6
P
(
High demand

air temperature is normal
) = 0
.
2
P
(
High demand

air temperature is above normal
) = 0
.
05
.
54
CHAPTER 5. CONDITIONAL PROBABILITY
55
We can calculate these conditional probabilities using the formula
P
(
A

B
) =
P
(
A
and
B
)
P
(
B
)
,
that is, in terms of the probability of both events occurring,
P
(
A
and
B
)
, and the probability of
the event that has already taken place,
P
(
B
)
.
To see how this formula works, let’s consider the class of students from Exercises 4 (on page
53).
Student
Height
Weight
Shoe
Student
Height
Weight
Shoe
Number
Sex
(m)
(kg)
Size
Number
Sex
(m)
(kg)
Size
1
M
1.91
70
11.0
10
M
1.78
76
8.5
2
F
1.73
89
6.5
11
M
1.88
64
9.0
3
M
1.73
73
7.0
12
M
1.88
83
9.0
4
M
1.63
54
8.0
13
M
1.70
55
8.0
5
F
1.73
58
6.5
14
M
1.76
57
8.0
6
M
1.70
60
8.0
15
M
1.78
60
8.0
7
M
1.82
76
10.0
16
F
1.52
45
3.5
8
M
1.67
54
7.5
17
M
1.80
67
7.5
9
F
1.55
47
4.0
18
M
1.92
83
12.0
Suppose we want the probability that a student chosen at random from this class will be female
given that the student’s shoe size is less than 8. We could simply find the proportion of students
with shoe sizes less than 8 who are female. There are 7 students with shoe sizes less than 8 and
4 of these are female. So
P
(
Female

Shoe size less than 8
) =
4
7
.
This probability can also be calculated using the above formula as follows:
P
(
Shoe size less than 8
)
=
7
18
,
P
(
Shoe size less than 8 and female
)
=
4
18
;
and so
P
(
Female

Shoe size less than 8
)
=
P
(
Shoe size less than 8 and female
)
P
(
Shoe size less than 8
)
=
4
/
18
7
/
18
=
4
7