Chapter 33
Probability
33.1 Introduction to probability
33.1.1 Probability
The probability of something happening is the likelihood or chance of it happening. Values of probability lie
between 0 and 1, where 0 represents an absolute impossibility and 1 represents an absolute certainty. The probability of an event happening usually lies somewhere
between these two extreme values and is expressed
as either a proper or decimal fraction. Examples of
probability are
that a length of copper wire
has zero resistance at 100
◦ C 0
that a fair, six-sided dice
will stop with a 3 upwards
1
6
or 0.1667
that a fair coin will land
with a head upwards
1
2
or 0.5
that a length of copper wire
has some resistance at 100 ◦ C 1
If p is the probability of an event happening and q is the
probability of the same event not happening, then the
total probability is p + q and is equal to unity, since it
is an absolute certainty that the event either will or will
not occur; i.e., p + q = 1.
Problem 1. Determine the probabilities of
selecting at random (a) a man and (b) a woman
from a crowd containing 20 men and 33 women
(a) The probability of selecting at random a man, p,
is given by the ratio
number of men
number in crowd
i.e.
p =
20
20 + 33
=
20
53
or 0.3774
(b) The probability of selecting at random a woman,
q, is given by the ratio
number of women
number in crowd
i.e.
q =
33
20 + 33
=
33
53
or 0.6226
(Check: the total probability should be equal to 1:
p =
20
53
and q =
33
53
, thus the total probability,
p + q =
20
53
+
33
53
= 1
hence no obvious error has been made.)
33.1.2 Expectation
The expectation, E, of an event happening is defined
in general terms as the product of the probability p of
an event happening and the number of attempts made,
n; i.e., E = pn.
Thus, since the probability of obtaining a 3 upwards
when rolling a fair dice is 1/6, the expectation of getting
a 3 upwards on four throws of the dice is
1
6
× 4, i.e.
2
3
Thus expectation is the average occurrence of an
event.
DOI: 10.1016/B978-1-85617-697-2.00033-8
Probability
33.1 Introduction to probability
33.1.1 Probability
The probability of something happening is the likelihood or chance of it happening. Values of probability lie
between 0 and 1, where 0 represents an absolute impossibility and 1 represents an absolute certainty. The probability of an event happening usually lies somewhere
between these two extreme values and is expressed
as either a proper or decimal fraction. Examples of
probability are
that a length of copper wire
has zero resistance at 100
◦ C 0
that a fair, six-sided dice
will stop with a 3 upwards
1
6
or 0.1667
that a fair coin will land
with a head upwards
1
2
or 0.5
that a length of copper wire
has some resistance at 100 ◦ C 1
If p is the probability of an event happening and q is the
probability of the same event not happening, then the
total probability is p + q and is equal to unity, since it
is an absolute certainty that the event either will or will
not occur; i.e., p + q = 1.
Problem 1. Determine the probabilities of
selecting at random (a) a man and (b) a woman
from a crowd containing 20 men and 33 women
(a) The probability of selecting at random a man, p,
is given by the ratio
number of men
number in crowd
i.e.
p =
20
20 + 33
=
20
53
or 0.3774
(b) The probability of selecting at random a woman,
q, is given by the ratio
number of women
number in crowd
i.e.
q =
33
20 + 33
=
33
53
or 0.6226
(Check: the total probability should be equal to 1:
p =
20
53
and q =
33
53
, thus the total probability,
p + q =
20
53
+
33
53
= 1
hence no obvious error has been made.)
33.1.2 Expectation
The expectation, E, of an event happening is defined
in general terms as the product of the probability p of
an event happening and the number of attempts made,
n; i.e., E = pn.
Thus, since the probability of obtaining a 3 upwards
when rolling a fair dice is 1/6, the expectation of getting
a 3 upwards on four throws of the dice is
1
6
× 4, i.e.
2
3
Thus expectation is the average occurrence of an
event.
DOI: 10.1016/B978-1-85617-697-2.00033-8