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Probability and Discrete Probability Distribution
The concept
• Probability is a numerical measure of the likelihood that an event will occur
• Event is a set of outcomes to which a probability is assigned
– An event is certain if it will always happen (e
...
: it is certain that a die will land on a number 1 – 6)
– An event is impossible if it will never happen(e
...
: it is impossible that a die will land on number 10)
• A certain event has a probability of 1 and an impossible event has a probability of 0
Sample space: the collection of all possible events (e
...
: all 6 faces of a die; all 52 cards in a deck of
cards)
• Simple event: an outcome from a sample space with one characteristic (e
...
: a red card from a deck of
cards)
• Joint event: events occurring simultaneously (e
...
: an ace that is also red from a deck of cards)
• Complement event: the complement of an event A (written A’) is the set of all outcomes in the
sample space that are not included in the outcomes of event A (e
...
: when tossing a coin, the
complement of a head is tail)
• Mutually exclusive event: events are mutually exclusive (or disjoint) if they cannot happen at the
same time (e
...
: when tossing a coin either a head or a tail might turn up, but not head and tail at the
same time)
Independent events: whatever happens in one event has absolutely nothing to do with what will
happen next because:
– the two events are unrelated
– you repeat an event with an item whose numbers will not change
– you repeat the same activity, but you replace the item that was removed

• Dependent event: what happens during the second event depends upon what happened before
(example: a box contains 3 white marbles and 4 black marbles
...
All possible events are
likely (example: heads and tails in a coin are collectively exhaustive)

1

Approaches

empirical

priori classical

classical

probability

probability
based on prior knowledge of

the probability of an event is

the process involved

based on observed data

e
...
the probability of
selecting

e
...
find the probability of
selecting a male taking

a face card from a standard

statistics from the population

deck of 52 cards

described

P = X/T = 12/52

event is determined by
an
individual, based on
that
subjective
probability

High in personal bias –
as
differs from person to
person

person’s past
experience,

& no formal
calculations and

personal opinion,
and/or

only reflect the
subject’s

analysis of a particular

opinions and past
experience

situation

2

e
...
Probability of
Sydney win a football
away match in MFL

Simple & Joint probability

RED (X)

F(x,y)
1
2
3
4
fY(y)

BLACK
(Y)
2
1/16
1/16
1/16
1/16
4/16

1
1/16
1/16
1/16
1/16
4/16

3
1/16
1/16
1/16
1/16
4/16

4
1/16
1/16
1/16
1/16
4/16

Fx(x)
4/16
4/16
4/16
4/16
1

Table 1 : BMI of Police officers in Kota Samarahan Police Department
Thin
BMI < 19

Male
Female
Total

Normal
BMI 19 – 25

Overweight
BMI >25

Total

21
5
26

32
19
51

60
21
81

113
45
158

Table 2: contingency table for BMI of Police officers in London Police
Department

Thin
BMI < 19

Normal
BMI 19 – 25

Overweight
BMI >25

Total

Male
0
...
20
0
...
72
Female
0
...
12
0
...
28
Total
1
0
...
32
0
...
13

Thin = 0
...
03

Male = 0
...
32
Female = 0
...
39
Overweight
=0
...
13

Probability Rules
• When there are 2 events (A,B) possibly to occurs, determine the possibility that
event A, B or C occurs separately or all occurs together
• If event A, and B are mutually exclusive (disjoint), then P(A or B) = P(A)+P(B),
otherwise P(A or B) = P(A)+P(B) – P(A ∩ B)
• E
...
: In a marketing study on identify student purchasing behavior in Oxford
University, random sampling is applied
...
Find the Probability that student picked from this group is either a
freshmen or sophomore
...
7002
6852

6852

Non mutually exclusive (joint) P(A or B) =
P(A)+P(B) – P(A ∩ B)
• Take example on study in KSPD, find the probability that
police officer picked for the new exercise program at
random is a overweight or male
...
8481
158
Multiplication rule used when to determine the probability that two
events A and B both occurs
...
g
...
Joel want to make apple
crumbles and will pick apples randomly from the bag
...

• Note that the color of the first apple picked will effect on probability of the
5

second picked
...
2

Bayes’ Theorem
• In the conditional probability:
– P (A|B) = P(A and B)/P(B)
• When rearranged:
– P(A and B) = P(A|B) P(B)
– P(B and A) = P(B|A) P(A)
• Therefore:
– P(A|B) P(B) = P(B|A) P(A)

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