TASK 4.1
Underlying terminology in Probability, Combing
events, Mutually Exclusive and Independent
events
1
Whether you are aware of it or not, chance plays an
important part in our lives. We have to make hundreds of
decisions every day, all of which involve an element of
uncertainty.
For instance, when crossing a road, we have to decide
whether to cross immediately or wait. We have to weigh up
the chance of reaching the other side safely. Or, when
having a shower, we have to decide whether the water is
too cold, too hot or just right.
INTRODUCTION
2
• Probability is the study of events that may or may not
happen rather than events that will or have already
happened.
• If you pursue your studies of statistics at a later stage, you
will find that knowledge of some aspects of probability is
essential.
• Probability can be used to measure the degree of
uncertainty or risk associated with the use of sample
information and to draw conclusions about corresponding
populations.
INTRODUCTION
4
In statistics, any activity or process that results in some
information being obtained is called an experiment.
For example,
• tossing a coin and observing whether tails or heads
results,
• determining whether a certain production process results
in defective or non-defective items,
• conducting a survey to ascertain the number of cars that
pass through a busy intersection every minute, and so on.
Experiments,
Outcomes and Trials
5
• The outcome of an experiment is the result which occurs
when the experiment is performed.
• When the outcome is uncertain, i.e., cannot be predicted
with certainty before the experiment is completed, the
experiment is called a random experiment.
• Each performance of the experiment is called a trial.
• If the outcome that occurs is the one we were interested
in obtaining, it is called a favourable outcome.
Experiments,
Outcomes and Trials
6
• The sample space is the set of all possible outcomes, and
is often denoted by S.
• One and only one of the outcomes will occur.
• For example, an event E consists of one or more
individual outcomes of the experiment and is thus any
subset of S.
• Note: Events are usually denoted by upper case letters
such as A, B, C, …. E1 , E2 , E3 , K
Sample Space and Events
7
• When a die is rolled, do you agree that one of the
outcomes 'a l ‘, 'a 2', 'a 3', 'a 4', 'a 5' or 'a 6' must occur?
• We say that these events are exhaustive since they
include all of the possible outcomes of the experiment.
Sample Space and Events
8
• In probability, we often use set theory notation to list the
individual outcomes in an event.
• To illustrate, suppose an experiment consists of rolling a
regular six-faced die (singular of dice) once.
• Since the die could show a 1, 2, 3, 4, 5 or 6, there are six
possible outcomes so the sample space is:
S = {1, 2, 3, 4, 5, 6}
Using Sets
9
• If A is the event 'rolling a 3 ', then:
A= {3}
• If B is the event 'an even number', then:
B = {2, 4, 6}
• If we roll the die and observe the outcome 'a 4' the event
B will have occurred, but the event A will not have
occurred.
• Note: The event A is an example of a simple event since
it consists of a single outcome. B is not a simple event.
Using Sets
10
• Suppose the die is rolled twice in succession ( or two
dice are rolled once). Do you agree that each of the six
possible outcomes obtained on the first roll can be
associated with the six possible outcomes obtained on the
second roll?
• The sample space thus consists of the
6 × 6 = 32 outcomes
S = { 1, 1
2, 1
3, 1
4, 1
5, 1
6, 1
,
,
,
,
,
,
Using Sets
1, 2
2, 2
3, 2
4, 2
5, 2
6, 2
,
,
,
,
,
,
1, 3
2, 3
3, 3
4, 3
5, 3
6, 3
,
,
,
,
,
,
1, 4
2, 4
3, 4
4, 4
5, 4
6, 4
,
,
,
,
,
,
1, 5
2, 5
3, 5
4, 5
5, 5
6, 5
,
,
,
,
,
,
1, 6
2, 6
3, 6
4, 6
5, 6
6, 6
,
,
,
,
,
}
11
S = { 1, 1
2, 1
3, 1
4, 1
5, 1
6, 1
,
,
,
,
,
,
1, 2
2, 2
3, 2
4, 2
5, 2
6, 2
,
,
,
,
,
,
1, 3
2, 3
3, 3
4, 3
5, 3
6, 3
,
,
,
,
,
,
1, 4
2, 4
3, 4
4, 4
5, 4
6, 4
,
,
,
,
,
,
1, 5
2, 5
3, 5
4, 5
5, 5
6, 5
,
,
,
,
,
,
1, 6
2, 6
3, 6
4, 6
5, 6
6, 6
,
,
,
,
,
}
(l, l) means
• l`s on both rolls,
(1, 2) means
• a l on the first roll and a 2 on the second roll, and so on.
• We can use a Venn diagram to represent the sample
space and other events for our experiment of rolling a die.
Using Sets
12
A= {3} and B = {2, 4, 6}
B
A
4
2
3
6
1
5
• In such a diagram, the sample space is represented by points inside a
rectangular region.
• Any other event is represented by points inside a circular (or other
shaped) region within the rectangle.
Using Sets
13
• As well as using a Venn diagram to represent a sample
space and events, we can often use a tree diagram as a
useful visual aid to determine the sample space for
certain experiments.
• The outcomes can be represented as a branch of a tree.
• To illustrate, consider the experiment of tossing a single
coin and observing the results.
Determining the Sample Space 14
T
H
• T denotes the event 'a tail' and H represent 'a head'.
• The sample space is:
S = {T, H}
• If A is the event 'a tail', then A= {T}.
Determining the Sample Space 15
• Tree diagrams are very useful when determining all of
the possible outcomes associated with a number of
successive ( or simultaneous) events, and as we shall see
later can also be used to calculate probabilities of
sequences of events.
• Trials involving only two possible outcomes, such as a
coin toss, are usually described as Bernoulli Trials.
• The following Tree diagram shows the outcomes when
two coins are tossed (or a coin is tossed twice).
Determining the Sample Space 16
Second
toss
The first two branches show
the possible results from the
first toss of the coin (T or H).
First
toss
Outcome
T
TT
H
TH
T
HT
H
HH
T
The next branches show the
results of the second toss.
The sample space is:
S = {TT, TH, HT, HH}
H
If B is the event 'one tail', then
B = {TH, HT}
Determining the Sample Space
17
Sometimes, we use the
subscript 1 to denote the
outcome from the first coin
and the subscript 2 for that
of the second, so the sample
space would be:
𝑆 = {T1 T2 , T1 H2 , H1 T2 , H2 T2 }
Now draw a tree diagram
for a three coins toss, to
find the sample space.
Second
toss
First
toss
Outcome
T
TT
H
TH
T
HT
H
HH
T
H
Determining the Sample Space 18
• S = {TTT, TTH, THT,
HTT, THH, HTH,
HHT, HHH}
If C is the event '2 tails',
then
• C = {TTH, THT, HTT}.
T
T
---- TTT
H
---- TTH
T
---- THT
H
---- THH
T
---- HTT
H
---- HTH
T
---- HHT
H
---- HHH
T
H
H
1ST toss
T
H
2nd toss
3rd toss
Determining the Sample Space 19
• With the sample space
of 3 coin tosses, an
interesting pattern starts
to emerge.
• If we sum the like
commutative terms, (i.e
TTH=THT=HTT, etc.)
we see:
• S = {1 x TTT, 3x TTH,
3 x HHT, 1 x HHH}
T
T
---- TTT
H
---- TTH
T
---- THT
H
---- THH
T
---- HTT
H
---- HTH
T
---- HHT
H
---- HHH
T
H
H
1ST toss
T
H
2nd toss
3rd toss
The Pascal Triangle
20
• If we express this in function form we get the following
recognisable polynomial expansion (cubic symmetric
polynomial)
𝑓 S = T 3 + 3T 2 H + 3TH2 + H3
• Examining the coefficients for the set S or for f(S) we get
the following coefficients
1, 3, 3, 1
The Pascal Triangle
21
• By looking at coefficients when expanding the tree, we can
readily discern the following pattern ...
1
1
1
1
1
1
5
2
3
4
1
1
3
1
6 4 1
10 10 5 1
• Each inner number is the addition of the 2 numbers above it.
The Pascal Triangle
22
• This triangular pattern was studied in-depth by Pascal and
is subsequently referred to as Pascal's Triangle. It has
application to Bernoulli trials, Binomial expansions and
many other fields in mathematics.
• It should also be noted that this type of expansion had
been either described or studied by other mathematicians
long before Pascal.
• One such notable person is Chinese mathematician Yang
Hui. As such, this triangular pattern is known as the Yang
Hui triangle in China.
The Pascal Triangle
23
• Because we will shortly be considering various
probability rules, which involve the complement,
intersection and union of events, we need to ensure that
we know how to use these set operations.
Combining Events
24
• The complement of an event A is the event A (read as 'not A') ,
which is the set of all possible outcomes in S that are not
included in A. Its Venn Diagram is ….
S
A
A
• If A is represented by the unshaded region, the shaded area
represents A .
Complementation
25
• For our experiment of rolling a single die where
S = { l, 2, 3, 4, 5, 6}
If A= {3},
Then the set 'not a 3’ is
A = {l, 2, 4, 5, 6}
Complementation
26
• If A and B are two events, the event A ∩ B (read as 'A and B’)
consists of all outcomes that belong to both A and B. It is the
intersection of A and B.
• This is a compound event since it is compounded from the
events A and B, as shown in the Venn Diagram…
S
A
B
• The joint event A ∩ B occurs only if the events A and B both
occur. It is represented by the overlap.
Intersection
27
• For an experiment of rolling a die where
S = {1, 2, 3, 4, 5, 6}
• With events E1 = {1, 2, 3} and E2 = { 2, 3, 4}, what
is E1 ∩ E2 ?
E1 ∩ E2 ={2, 3}
Intersection
28
• If A and B are two events, the event A ∪ B (read as ‘A or B’)
consists of all outcomes that belong to either A, or B, or both.
It is the union of A and B.
• This is a compound event since it is compounded from the
events A or B, as shown in the Venn Diagram…
S
A
B
• The event A ∪ B occurs if either A or B occurs ( or both).
Union
29
• For an experiment of rolling a die where
S = {1, 2, 3, 4, 5, 6}
• With events E1 = {1, 2, 3} and E2 = { 2, 3, 4}, what
is E1 ∪ E2 ?
E1 ∪ E2 ={1, 2, 3, 4}
Intersection
30
• Two events A and B are said to be mutually exclusive or
disjoint if the occurrence of one precludes the occurrence of
the other. The events have no outcomes in common and do
not overlap.
S
A
B
• Since A ∩ B = ϕ (the empty set), it is impossible for both A
and B to occur simultaneously.
Mutually Exclusive Events
31
• As a further illustration, suppose two student committee
members are standing for president.
If A is the event 'Jay is elected’
and B is the event 'Lee is elected’,
• then A and B are mutually exclusive events since both
cannot be elected as president.
Mutually Exclusive Events
32
• Two events are said to be independent if the events in no
way affect each other.
• Two events that are related so that the occurrence of one
is influenced by that of the other are not independent and
are said to be dependent events.
• We can sometimes use the context of a problem to infer
whether the events are independent or not.
Independent Events
33
In general, two physically different events will be
independent. For example:
• being a woman and being a vice-chancellor
• interest rates and the weather
• walking to work and wearing brown shoes
whereas the following pairs of events would not be
independent
• amount of time spent studying and passing a subject
• housing sales and mortgage interest rates
• smoking cigarettes and developing lung cancer.
Independent Events
34
• In games of chance, the outcomes when coins are tossed
are independent since the outcome of one toss should
have no effect on the outcome of any other toss
(assuming, of course, that the coins are fair or unbiased).
• Drawing cards from a pack of cards without replacing
the previous card drawn is an Example of dependent
events.
• If the cards are replaced after being drawn, the events
would be independent.
Independent Events
35
• Do not fall into the common trap of thinking that the
terms 'mutually exclusive' and 'independent' are the
same.
• Suppose that A and B are two mutually exclusive events
and that event B has occurred. Then, the event A cannot
occur.
• This means that the occurrence of B does influence the
occurrence of A. That is, the mutually exclusive events
are dependent events.
Independent Events
36
• The difference between the two concepts can be
illustrated fairly simply. Suppose a coin is tossed twice.
Define the events:
T1 = first toss ( of coin) results in a tail
• and so on. Then, do you agree that (for example):
T1 , H1 are mutually exclusive
but T1 , T2 are independent?
Independent Events
37
Exercise Questions
Q1) – Q5)
38
Consider the experiment where a single ticket is drawn
from a box containing ten tickets numbered 1, 2, 3, ….10.
a) List the sample space S for this experiment
S= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
b) List the subsets that correspond to the events
i) A: a number greater than 6
A(>6) = {7, 8, 9, 10}
ii) B: an odd number
B(odd) = {1, 3, 5, 7, 9}
Q1)
39
S
A
B
1
8
2
7
3
10
9
5
4
6
A  {1, 2, 3, 4, 5, 6}
Q1)b)iii) A
40
S
A
B
1
8
2
7
3
10
9
5
4
6
Since B is the event " not an odd number " or " an even number "
B  {2, 4, 6, 8, 10}, so A  B = {7, 8, 9, 10}  {2, 4, 6, 8, 10}={8, 10}
Q1)b)iv) A  B
41
S
A
B
1
8
2
7
3
10
9
5
4
6
A  B = {1, 2, 3, 4, 5, 6}  {1, 3, 5, 7, 9}
={1, 2, 3, 4, 5, 6, 7, 9}
Q1)b)v) A  B
42
a) Use a tree diagram to determine the sample space for the
children in three-child families
Q2)
43
G
---- GGG
B
---- GGB
G
---- GBG
B
---- GBB
G
---- BGG
B
---- BGB
B
G
---- BBG
2nd child
B
---- BBB
G
G
B
B
1ST child
G
3rd child
Q2)a)
44
S={GGG, GGB, GBG, GBB, BGG, BGB, BBG, BBB}
b) List the subsets corresponding to the events:
i) A: the youngest child is a boy
Youngest is a boy means the third child must be a boy, so
A = {GGB, BGB, GBB, BBB
ii) B: there is one girl
B = {GBB, BGB, BBG}
iii) C: no boy has an older sister
C = {BGG, BBG, BBB}
iv) D: at least one boy
Q2)
At least one boy means one or more boys, i.e. one, two or three so,
D = {GGB, GBG, GBB, BGG, BGB, BBG, BBB}
45
b) List the subsets corresponding to the events:
v) A  B
A∪B = {GGB, BGB, GBB, BBB} ∪ {GBB, BGB, BBG} =
{GGB, GBB, BGB, BBG, BBB}
vi) C  D
C∩D = {BGG, BBG,BBB} ∩
{GGB, GBG, GBB, BGG, BGB, BBG, BBB} =
{BBG, BGG, BBB} = C
Q2)
46
What is the compliment of:
a) The rate of inflation which will be less than 3% next
year?
The rate of inflation will be 3% or more next year
b) A family with 2 boys?
A family with children does not have 2 boys (which
means there can be less than 2 boys or 3 or more
boys)
Q3)
47
Are the following event mutually exclusive? Explain.
a) Being under 12 years old and being an adult.
Mutually exclusive. A parliamentarian must be at least 18 years old
b) Throwing 2 dice. A is the event ‘a 6’ and B is event ‘a 4’.
Not Mutually exclusive. Can roll (4, 6) or (6, 4)
c) Throwing 2 dice. A is the event ‘a sum of 10’ and B is the event ‘a 6’.
Not Mutually exclusive. 10 can equal (6, 4) or (4, 6)
d) Drawing 2 cards. A is the event ‘a Jack’ and B is the event ‘a spade’.
Not Mutually exclusive. A card can be both a Spade and a Jack
e) Drawing 2 cards. A is the event ‘a spade’ and B is the event ‘a red card’.
Mutually exclusive. A card cannot be red and also a Spade
Q4)
48
Are the following events independent? Explain
a) Living in Australia and being a stamp collector.
Independent. Australians can be stamp collectors (or vice-versa)
b) The gender of your first child.
Independent. Birth of girl or boy does not affect gender of next birth
c) Getting an even number on one roll of a die and 5 on the second.
Independent. Outcome of 1st roll has no affect on 2nd roll
d) Drawing 2 cards at the same time and getting two clubs.
Not Independent. One club drawn reduces no. of clubs for 2nd card
e) Drawing a card, replacing it, drawing another card and getting two clubs.
Independent. By replacing a card, the number of clubs remains the
same for both draws
Q5)
49