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PROGRAMME 28
PROBABILITY
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 1
Introduction
Empirical probability
Classical probability
Certain and impossible events
Mutually exclusive and mutually non-exclusive events
Addition laws of probability
Independent events and dependent events
Multiplication law of probabilities
Conditional probability
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Introduction
Empirical probability
Classical probability
Certain and impossible events
Mutually exclusive and mutually non-exclusive events
Addition laws of probability
Independent events and dependent events
Multiplication law of probabilities
Conditional probability
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Introduction
Notation
Types of probability
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Introduction
Notation
Every trial or experiment has one or more possible outcomes. For example,
rolling a six-sided die is a trial with six possible outcomes. An event, denoted
by A, is a collection of one or more of those outcomes. For example, throwing
an even number in the roll of a six-sided die is an event which consists of three
outcomes.
The probability, denoted by P(A) of an event A is a measure of the likelihood
of the event occurring in any one trial or experiment.
The probabilities of the various events associated with a trial can be defined in
one of two ways.
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Introduction
Types of probability
The probabilities of the various events associated with a trial can be defined
either:
(a) Empirically: by repeating the experiment a number of times and noting the
relative frequencies of the events.
(b) Classically: by defining the probabilities beforehand based on a knowledge
of the experiment and its possible outcomes.
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 1
Introduction
Empirical probability
Classical probability
Certain and impossible events
Mutually exclusive and mutually non-exclusive events
Addition laws of probability
Independent events and dependent events
Multiplication law of probabilities
Conditional probability
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Empirical probability
Expectation
Success or failure
Multiple samples
Experiment
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Worked examples and exercises are in the text
Programme 28: Probability
Empirical probability
Empirical probability is based on previous known results. The relative
frequency of the number of times an event has previously occurred is taken as
the indication of its likely occurrence in the future
For example: A random batch of 240 components is inspected and 20 are
found to be defective. Therefore, if one component is picked at random from
this batch, the chance of its being defective is 20 in 240. that is 1 in 12.
In this case if A = {defective component} then the probability of selecting a
defective component is:
1
P ( A) 
12
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Empirical probability
Expectation
The expectation E of an event A occurring in N trials is defined as the product
of the probability of A occurring and the number of trials:
E  N  P( A)
For example, in a production run of 600 components where the probability of
any one component being defective is 1/12 the expectation is that there will
be:
1
600   50
12
defectives in the run.
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Empirical probability
Success or failure
If, in a trial, a particular event A does occur that can be recorded as a success.
If the event does not occur that can be recorded as a failure. In N trials there
will be x successes and N – x failures. Now:
x  ( N  x)  N
so that:
Hence, since:
x Nx

1
N
N
P({success}) 
then:
STROUD
x
Nx
and P({failure}) 
N
N
P({success})  P({failure})  1
Worked examples and exercises are in the text
Programme 28: Probability
Empirical probability
Success or failure
Denoting by A the event {success} then notA is the event {failure} then:
P( A)  P(notA)  1
The event notA is denoted by the complement of A:
A
That is:
 
P  A  P A  1
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Empirical probability
Multiple samples
A single sample of size n taken from a total population of size N is not
necessarily representative of the population.
Another sample of size n could well display different properties.
However, continued sampling will provide a cumulative effect that does
demonstrate consistency.
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Empirical probability
Experiment
(a) Take a deck of 52 playing cards, shuffle well and deal out 12 cards (n) at
random.
(b) Count the number of spades (x)
(c) Replace the sample, re-shuffle and repeat the process.
(d) Calculate the average number of spades in the two samples – the running
average.
In this way a table can be constructed:
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Empirical probability
Experiment
The results from one such experiment gave the following distribution of
spades:
Number of spades in 40 trials of a 12-card sample
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Worked examples and exercises are in the text
Programme 28: Probability
Empirical probability
Experiment
Plotting the running average (cum x)/r against r gave the following graph:
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Worked examples and exercises are in the text
Programme 28: Probability
Empirical probability
Experiment
From the graph it is seen that the running average is settling down to a value
of 3 spades in a sample of 12 cards.
Since there are 13 spades in a deck of 52 playing cards there is an expectation
that there will be:
13
 12  3
52
Spades in a sample of 12 – but this is taking us towards the second way of
defining probabilities.
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 1
Introduction
Empirical probability
Classical probability
Certain and impossible events
Mutually exclusive and mutually non-exclusive events
Addition laws of probability
Independent events and dependent events
Multiplication law of probabilities
Conditional probability
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Classical probability
The classical approach is to consider the number of ways that an event could
possibly occur and take the ratio of that to the total number of possible
outcomes:
P( A) 
number of ways in which event A can occur
total number of possible outcomes
Since there are 13 spades in a deck of 52 playing cards, the probability of a
card drawn at random being a spade is then:
P({spade}) 
STROUD
13 1

52 4
Worked examples and exercises are in the text
Programme 28: Probability
PART 1
Introduction
Empirical probability
Classical probability
Certain and impossible events
Mutually exclusive and mutually non-exclusive events
Addition laws of probability
Independent events and dependent events
Multiplication law of probabilities
Conditional probability
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Certain and impossible events
Given that an event A can occur x times in n trials then:
x
nx
and so P A 
n
n
If event A is certain to occur then x = n, in which case:
P  A 
P  A 
 
n
nn
=1 and so P A 
0
n
n
 
The probability of certainty is 1 and the probability of impossibility is 0.
Therefore, for any event:
0  P  A  1
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 1
Introduction
Empirical probability
Classical probability
Certain and impossible events
Mutually exclusive and mutually non-exclusive events
Addition laws of probability
Independent events and dependent events
Multiplication law of probabilities
Conditional probability
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Mutually exclusive and mutually non-exclusive events
Mutually exclusive events are events that cannot occur together. For example,
in rolling a six-sided die a 5 and a 6 cannot be rolled at the same time in any
one trial.
Mutually non-exclusive events are events that can occur simultaneously. For
example, in rolling a six-sided die a 6 and an even number can be rolled at the
same time in any one trial
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 1
Introduction
Empirical probability
Classical probability
Certain and impossible events
Mutually exclusive and mutually non-exclusive events
Addition laws of probability
Independent events and dependent events
Multiplication law of probabilities
Conditional probability
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Addition laws of probability
If there are n possible outcomes to a trial, of which x give an event A and y
give an event B then if:
(a) A and B are mutually exclusive events, then:
P  A or B   P  A  P  B 
(b) A and B are mutually non-exclusive events, then:
P  A or B   P  A  P  B   P  A and B 
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 1
Introduction
Empirical probability
Classical probability
Certain and impossible events
Mutually exclusive and mutually non-exclusive events
Addition laws of probability
Independent events and dependent events
Multiplication law of probabilities
Conditional probability
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Independent events and dependent events
Events are independent when the occurrence on one event does not affect the
probability of the occurrence of the second event.
For example, drawing one card from a deck and then drawing a second card
after the first card has been replaced are independent events.
Events are dependent when the occurrence on one event does affect the
probability of the occurrence of the second event.
For example, drawing one card from a deck and then drawing a second card
after the first card has not been replaced are dependent events.
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 1
Introduction
Empirical probability
Classical probability
Certain and impossible events
Mutually exclusive and mutually non-exclusive events
Addition laws of probability
Independent events and dependent events
Multiplication law of probabilities
Conditional probability
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Multiplication law of probabilities
If events A and B are independent events then:
P  A and B   P  A  P  B 
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 1
Introduction
Empirical probability
Classical probability
Certain and impossible events
Mutually exclusive and mutually non-exclusive events
Addition laws of probability
Independent events and dependent events
Multiplication law of probabilities
Conditional probability
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Conditional probability
The probability of event B occurring given that event A has already occurred is
denoted by:
P  B A
If A and B are independent events the prior occurrence of event A will have no
effect on the probability of the occurrence of B and so:
P  B A  P  B 
If A and B are dependent events the prior occurrence of event A will have an
effect on the probability of the occurrence of B and in this case:
P  A and B   P  A  P  B A
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 2
Discrete probability distribution
Permutations and combinations
General binomial distribution
Mean and standard deviation of a probability distribution
The Poisson probability distribution
Continuous probability distributions
Standard normal curve
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 2
Discrete probability distribution
Permutations and combinations
General binomial distribution
Mean and standard deviation of a probability distribution
The Poisson probability distribution
Continuous probability distributions
Standard normal curve
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Discrete probability distribution
A coin is repeatedly tossed and the possible outcomes are listed:
1 trial:
T
H
2 trials:
TT
TH
HT
HH
3 trials:
TTT
TTH
THT
HTT
THH
HTH
HHT
HHH
4 trials
...
...
...
...
...
...
...
...
...
The results are then tabulated and the probabilities of tossing a head are listed:
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Discrete probability distribution
There is clearly a pattern here. The probabilities are the separate terms of the
binomial expansion:
n
1 1
  
2 2
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Discrete probability distribution
Recall that the binomial coefficients are given by Pascal’s triangle:
Further progress now requires a knowledge of permutations and combinations.
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 2
Discrete probability distribution
Permutations and combinations
General binomial distribution
Mean and standard deviation of a probability distribution
The Poisson probability distribution
Continuous probability distributions
Standard normal curve
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Permutations and combinations
Permutations
Combinations
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Worked examples and exercises are in the text
Programme 28: Probability
Permutations and combinations
Permutations
Given the three letters A, B and C they can be arranged in 6 different ways:
AB, AC, BC, BA, CA, CB
Each arrangement of 2 letters is called a permutation and the permutations of
2 items out of 3 is denoted by:
3
P2
In this case it is seen that:
3
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P2  6
Worked examples and exercises are in the text
Programme 28: Probability
Permutations and combinations
Permutations
Given n different objects taken in arrangements of r at a time, the first one can
be selected in n ways, the second in n – 1 ways, the third in n – 2 ways and so
on until the rth can be selected in n – r + 1 ways to give:
n
Pr  n(n  1)(n  2)( )(n  r  1) permutations
Now:
n
Pr 
n(n  1)(n  2)( )(n  r  1)(n  r )( )(3)(2)(1)
(n  r )( )(3)(2)(1)
That is:
n
STROUD
Pr 
n!
(n  r )!
Worked examples and exercises are in the text
Programme 28: Probability
Permutations and combinations
Combinations
Given the three letters A, B and C they can be selected in 3 different ways:
AB, AC, BC (note that BA = AB; they are the same selection)
Each selection of 2 letters is called a combination and the combinations of 2
out of 3 is denoted by:
3
C2
In this case it is seen that:
C2  3
3
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Permutations and combinations
Combinations
Given n different objects taken in selections of r at a time, the first one can be
selected in n ways, the second in n – 1 ways, the third in n – 2 ways and so on
until the rth can be selected in n – r + 1 ways to give:
n
Pr  n(n  1)(n  2)( )(n  r  1) permutations
Each selection can be rearranged in r! different ways to give r! different
permutations but still remain the same selection. Therefore there are:
n
Cr 
n!
(n  r )!r !
combinations.
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 2
Discrete probability distribution
Permutations and combinations
General binomial distribution
Mean and standard deviation of a probability distribution
The Poisson probability distribution
Continuous probability distributions
Standard normal curve
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
General binomial distribution
If a trial has two possible outcomes, success with a probability of p and failure
with a probability of q then:
p+q=1
In a sequence of n trials the probability of r successes is given by the general
binomial term:
P({r successes}) 
n!
q nr p r
(n  r )!r !
This distribution of probabilities is called the binomial probability distribution.
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
General binomial distribution
For a binomial probability distribution:
P({r successes}) 
n!
q nr p r
(n  r )!r !
where n = 5, p = 0.2 and therefore q = 0.8 the following table can be
constructed:
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
General binomial distribution
These values can then be displayed as a probability histogram:
(a) The probability of any particular outcome is given by the height of each column,
but since the columns are 1 unit wide this equates to the area of each column.
(b) The total probability is 1 so:
the total area of the probability histogram is 1
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 2
Discrete probability distribution
Permutations and combinations
General binomial distribution
Mean and standard deviation of a probability distribution
The Poisson probability distribution
Continuous probability distributions
Standard normal curve
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Mean and standard deviation of a probability distribution
There are two very simple and useful formulas for the mean and the standard
deviation of a binomial probability distribution:
Mean   np
Standard deviation   npq
where:
n  number of possible outcomes in any single trial
p  probability of success in any single trial
q  probability of failure in any single trial
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 2
Discrete probability distribution
Permutations and combinations
General binomial distribution
Mean and standard deviation of a probability distribution
The Poisson probability distribution
Continuous probability distributions
Standard normal curve
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
The Poisson probability distribution
When the number of trials is large n ≥ 50 and the probability of success is
small p  0.1 the binomial probability distribution can be closely
approximated by the Poisson probability distribution.
For a sequence of n trials the Poisson probability of r successes is given as:
e   r
P(r ) 
r!
where:
  np
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 2
Discrete probability distribution
Permutations and combinations
General binomial distribution
Mean and standard deviation of a probability distribution
The Poisson probability distribution
Continuous probability distributions
Standard normal curve
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Continuous probability distributions
Normal distribution curve (or normal curve)
The binomial and Poisson probability distributions refer to discrete events.
Where continuous variables are involved the concern is one of finding the
probability that a particular variable lies between certain limiting values.
For this reference is made to a continuous probability distribution – the normal
probability distribution.
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Continuous probability distributions
Normal distribution curve (or normal curve)
The normal probability distribution curve has the equation:
1
 ( x   )2 /  2
1
y
e 2
 2
Which has the graph of the bell shaped curve with mean  and standard
deviation 
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
PART 2
Discrete probability distribution
Permutations and combinations
General binomial distribution
Mean and standard deviation of a probability distribution
The Poisson probability distribution
Continuous probability distributions
Standard normal curve
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Standard normal curve
In practice it is not convenient to deal with the normal distribution so it is
converted into the standard normal distribution by defining the variable z as:
z
x

the standard normal variable
The equation for the normal distribution then becomes:
y   ( z) 
1  z2 / 2
e
2
with mean 0 and standard deviation 1.
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Standard normal curve
Note the following:
(a)
(b)
(c)
(d)
Mean  = 0
The z-values are in standard deviation units
Total area under the curve = 1
Area between z = a and z = b represents the probability P(a  z  b)
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Standard normal curve
Similarly, the probability of a randomly selected value of z where 0.5  z  1.5
is given by the area shaded:
That is:
1.5
P(0.5  z  1.5) 

z  0.5
1  z2 / 2
e
dz
2
This integral cannot be evaluated
by ordinary means, so a table is
used giving the area under the
standard normal curve from
z = 0 to z = z1
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Standard normal curve
Area under the standard normal curve
STROUD
Worked examples and exercises are in the text
Programme 28: Probability
Learning outcomes
Understand the nature of probability as a measure of chance
Compute expectations of events from an experiment with a number of outcomes
Assign classical measures to the probability and be able to define the probabilities of
both certainty and impossibility
Distinguish between mutually exclusive and mutually non-exclusive events and
compute their probabilities
Compute conditional probabilities
Evaluate permutations and combinations
Use the binomial and Poisson probability distributions to calculate probabilities
Use the standard normal probability distribution to calculate probabilities
STROUD
Worked examples and exercises are in the text