Chapter 56
Probability
56.1 Introduction to 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
either as 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 does or does
not occur, i.e. p + q = 1
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.
Dependent event
A dependent event is one in which the probability of
an event happening affects the probability of another
event happening. Let 5 transistors be taken at random
from a batch of 100 transistors for test purposes, and the
probability of there being a defective transistor, p 1 , be
determined. At some later time, let another 5 transistors
be taken at random from the 95 remaining transistors in
the batch and the probability of there being a defective
transistor, p 2 , be determined. The value of p 2 is different
from p 1 since batch size has effectively altered from 100
to 95, i.e. probability p 2 is dependent on probability p 1 .
Since 5 transistors are drawn, and then another 5 transistors drawn without replacing the first 5, the second
random selection is said to be without replacement.
Independent event
An independent event is one in which the probability
of an event happening does not affect the probability
of another event happening. If 5 transistors are taken at
random from a batch of transistors and the probability of
a defective transistor p 1 is determined and the process
is repeated after the original 5 have been replaced in
the batch to give p 2 , then p 1 is equal to p 2 . Since the
5 transistors are replaced between draws, the second
selection is said to be with replacement.
Conditional probability
Conditional probability is concerned with the probability of say event B occurring, given that event A has
already taken place.
If A and B are independent events, then the fact that
event A has already occurred will not affect the probability of event B.
If A and B are dependent events, then event A having
occurred will effect the probability of event B.
Probability
56.1 Introduction to 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
either as 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 does or does
not occur, i.e. p + q = 1
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.
Dependent event
A dependent event is one in which the probability of
an event happening affects the probability of another
event happening. Let 5 transistors be taken at random
from a batch of 100 transistors for test purposes, and the
probability of there being a defective transistor, p 1 , be
determined. At some later time, let another 5 transistors
be taken at random from the 95 remaining transistors in
the batch and the probability of there being a defective
transistor, p 2 , be determined. The value of p 2 is different
from p 1 since batch size has effectively altered from 100
to 95, i.e. probability p 2 is dependent on probability p 1 .
Since 5 transistors are drawn, and then another 5 transistors drawn without replacing the first 5, the second
random selection is said to be without replacement.
Independent event
An independent event is one in which the probability
of an event happening does not affect the probability
of another event happening. If 5 transistors are taken at
random from a batch of transistors and the probability of
a defective transistor p 1 is determined and the process
is repeated after the original 5 have been replaced in
the batch to give p 2 , then p 1 is equal to p 2 . Since the
5 transistors are replaced between draws, the second
selection is said to be with replacement.
Conditional probability
Conditional probability is concerned with the probability of say event B occurring, given that event A has
already taken place.
If A and B are independent events, then the fact that
event A has already occurred will not affect the probability of event B.
If A and B are dependent events, then event A having
occurred will effect the probability of event B.