Download SSF1063: Statistics for Social Sciences

SSF1063: Statistics for
Social Sciences
LU4: Introduction to Probability
28th January 2008
Experiment, Outcomes & Sample Space

Definition
1. Experiment – a process that, when performed
results in one and only one of many observations
2. Observations – are called the outcomes of the
experiment
3. Sample space – The collection of all outcomes for
an experiment

A sample space is denoted by S. The sample space
for the example of inspecting a tennis ball is written
as
S = {good, defective}
Experiment, Outcomes & Sample Space

Examples of Experiment, Outcomes & Sample
Space
Experiment
Outcomes
Sample space
Toss a coin once
Head, tail
S = {Head, tail}
Take a test
Pass, fail
S = {Pass, fail}
Toss a coin twice
HH, HT, TH, TT
S = {HH, HT, TH, TT}
Roll a dice once
1, 2, 3, 4, 5, 6
S = {1, 2, 3, 4, 5, 6}
Play lottery
Win, lose
S = {Win, lose}
The Venn and Tree diagram

(a) Venn diagram and (b) tree diagram for two tosses
of a coin
1
2
H
HH
TH
HT
HH
H
T
HT
H
TT
TH
T
T
TT
The Venn and Tree diagram
Test yourself

Suppose we randomly select two persons from the
member of a club and observe whether the person
selected each time is a man or a woman. Write all
the outcomes for this experiment. Draw the Venn
and tree diagram for this experiment.
Simple and Compound Events

Event


Simple Event


A collection of one or more of the outcomes of an
experiment
An event that has only ONE outcome
Compound Event

Collection of MORE THAN ONE outcome for an
experiment
Simple and Compound Events

In a group of people, some are in favor of genetic
engineering and others are against it. Two persons are
selected at random from this group and asked whether they
are in favor of or against genetic engineering. How many
distinct outcomes are possible?
Draw a Venn diagram and a tree diagram for this
experiment. List all outcomes included in each of the
following events and mention whether they are simple of
compound events.
1. Both persons are in favor of genetic engineering
2. At most one person is against genetic engineering
3. Exactly one person is in favor of genetic engineering
Calculating Probability

Probability is a numerical measure of a likelihood
that a specific event will occur

The probability of an event always lie in the range 0 to 1




0 ≤ P(Ei) ≤ 1
0 ≤ P(A) ≤ 1
An event that cannot occur has zero probability –
IMPOSSIBLE EVENT; P(V) = 0
An event that is certain to occur has a probability equal
to 1 – SURE EVENT; P(S) = 1
Calculating Probability


Probability – numerical measure of a likelihood that a
specific event will occur
Property #1: The probability of an event always lie
in the range 0 to 1




0 ≤ P(Ei) ≤ 1  Simple event
0 ≤ P(A) ≤ 1  Compound event
An event that cannot occur has zero probability –
IMPOSSIBLE EVENT; P(V) = 0
An event that is certain to occur has a probability
equal to 1 – SURE EVENT; P(S) = 1
Calculating Probability

Property #2: The sum of all probabilities of all
events (or final outcomes) for an experiment, is
always 1.




ΣP(Ei) = P(E1) + P(E2) + P(E3) + … = 1
From this property, for the experiment of one toss
of a coin, P(H) + P(T) = 1
Experiment of two tosses of a coin;
P(HH) + P(HT) + P(TT) + P(TH) = 1
For one game of football by a professional team;
P(Win) + P(Loss) + P(Tie) = 1
Three Conceptual Approaches to
Probability
1.
2.
3.
Classical Probability
The relative frequency concept of
probability
The subjective probability concept
Classical Probability

According to the classical probability rule, the probability of
a simple event is equal to 1 divided by the total number of
outcomes for the experiment.
P(Ei) =

P(A) =
1
Total number of outcomes for the experiment
Number of outcomes favorable to A
Total number of outcomes for the experiment
Classical Probability

Find the probability of obtaining a head
and the probability of obtaining a tail for
one toss of a coin.



The outcomes (head or tail) are equally alike
P(head) =
1
= 1
Total number of outcomes
2
= 0.5
P(tail) = ?
Classical Probability


Find the probability of obtaining an even
number in one roll of a dice.
In a group of 500 women, 120 have
played golf at least once. Suppose one of
these 500 women is randomly selected.
What is the probability that she has played
golf at least once?
Relative Frequency Concept of Probability




Some events cannot be computed using the
classical probability rule because the outcomes
are not equal.
Different experiments produce different outcomes
Relative frequency used as an approximation for
the probability of that event.
If an experiment is repeated n times and an
event A is observed f times, according to the
relative frequency concept of probability,
P(A) = f
n
Relative Frequency Concept of Probability



Ten of the 500 randomly selected cars
manufactured at a certain auto factory are found
to be lemons. Assuming that the lemons are
manufactured randomly, what is the probability
that the next car manufactured at this auto
factory is a lemon?
Lets denote the total number of cars in the
sample and f the number of lemons in n.
Therefore, n = 500 and f = 10
Relative Frequency Concept of Probability


Using the relative frequency of lemons in 500
cars, we obtain:
P(next car is a lemon) = f/n = 10/500
= 0.02
This probability is actually the relative frequency
of lemons in 500 cars.
Frequency & Relative Frequency Distribution for the Sample
of Cars
Car
f
Relative Frequency
Good
Lemon
490
10
n = 500
490/500 = 0.98
10/500 = 0.02
Sum = 1.0
Relative Frequency Concept of Probability
Frequency & Relative Frequency Distribution for the
Sample of Cars
Car
Good
Lemon
f
Relative Frequency
490
10
490/500 = 0.98
10/500 = 0.02
n = 500

Sum = 1.0
Thus, from the relative frequency column,
P(next car is a lemon) = 0.02
P(next car is a good car) = 0.98
Subjective Probability

Probability assigned to an event based on
subjective judgment, experience,
information and belief.
1.The probability of you scoring 10% in
Test 1 scheduled tonight.
2. The probability of the interest rate to rise
by September 2007.
Counting Rule


Not all experiments deal with few
outcomes, that are easy to list.
Sometimes we might deal with large
number of outcomes, and it is uneasy to
list all the outcomes.
Therefore we need counting rules to find
the total number of outcomes.
Counting Rule to Find Total Outcomes


If an experiment consists of 3 steps and if
the first step can result in m outcomes,
the second step in n outcomes and the
third step in k outcomes, then:
Total outcomes for the experiment =
m·n·k
Counting Rule to Find Total Outcomes
Example 1:
 Suppose we toss a coin three times. This
experiment has 3 steps, first toss, second
toss and third toss. Each step has two
outcomes: a head and a tail. Thus,
 Total outcomes for three tosses of a coin
=2·2·2=8
Counting Rule to Find Total Outcomes
Example 2:
 A prospective car buyer can choose
between a fixed and a variable interest
rate and can also choose a payment
period of 36 months, 48 months or 60
months. How many total outcomes are
possible? And what are the outcomes?
Marginal Probabilities

Probability of a single event without consideration
of any other event. Also known as simple
probability
In favor
Male
Female
Total
Against
Total
15
45
60
4
36
40
19
81
100
Calculate the following:
i. P(in favor)
ii. P(against)
iii. P(male)
iv. P(female)
Conditional Probabilities

What is the probability that the selected male
employee is in favor of paying high salaries to
CEO?
In favor
Male
Female
Total
Against
Total
15
45
60
4
36
40
19
81
100
P(in favor|male) is read
as “the probability of the
employee selected is in
favor given that this
employee is a male”
P(in favor|male) = Number of male who are in favor
Total no. of male
Conditional Probabilities

What is the probability that the selected
employee in such favor is a female?
In favor
Male
Female
Total
Against
Total
15
45
60
4
36
40
19
81
100
P(female|in favor) is
read as…
Construct a tree
diagram to solve this
question.
P(female|in favor) = Number of female who are in favor
Total no. of staff who are in favor
Mutually Exclusive Events

Events that cannot occur together in one
repetition of an experiment
Consider the following events for one roll of a
die:
 A = an even number is observed = {2, 4, 6}
 B = an odd number is observed = {1, 3, 5}
 C = a number less than 5 is observed =
{1, 2, 3, 4, 5}
Compare A and B, then compare A and C.

Mutually Exclusive Events

Consider the following events:
1.Y = this student has eaten at Keranji Food
Court
N = this student has never eaten at Keranji
Food Court
2.X = household income is at least RM1200
Y = household income is more than RM900
Independent vs Dependent Events


Two events are considered independent when
the occurrence of one event does not affect the
probability of the occurrence of the other.
Either P(A|B) = P(A) or P(B|A) = P(B)
Dependent events on the other hand,
P(A|B) ≠ P(A) or P(B|A) ≠ P(B)
Independent vs Dependent Events
Defective (D) Good (G)
Machine A
Machine B
Total
Calculate P(D).
What is P(D|A)?
What is P(D|B)?
9
6
15
Total
51
34
85
Calculate P(G)
What is P(G|A)?
What is P(G|B)?
60
40
100
Independent vs Dependent Events
In favor
Male
Female
Total
Calculate P(In favor).
What is P(In Favor|F)?
What is P(In Favor|M)?
Against
Total
15
45
60
4
36
40
19
81
100
Calculate P(Against)
What is P(Against|F)?
What is P(Against|M)?
Complimentary Events


Events that are complement to one another,
Ā (read as “A bar”) includes all outcomes for an
experiment that are not in A
S
Ā
Ā
A
Venn diagram of
TWO complimentary
events
Complimentary Events


Therefore,
P(A) + P(Ā) = 1;
 P(A) = 1 - P(Ā) and
P(Ā) = 1 - P(A)
Thus, if we know the probability of an event, we
can find the probability of its complimentary
event by subtracting the given probability from 1
Complimentary Events
Example



In a group of 5000 adults, 3500 are in favor of
stricter gun control laws, 1200 are against such
laws and 300 have no opinion. One adult is
randomly selected from this group. Let A be the
event that this adult is in favor of stricter gun
control laws.
What is the complementary event of A?
What are the probabilities of the two events?
Joint Probability

The probability of the intersection of two
events
S
A
A
Āand
B
B
Intersection of
A and B
Joint Probability


Is given the multiplication rule:
P(A and B) = P(A)·P(B|A)
Can also denoted by P(A
B) or P(AB)
Joint Probability
College
Graduate (G)
Total
Male (M)
7
20
27
Female (F)
4
9
13
11
29
40
Total

Non College
Graduate (N)
What is the probability of selecting an
employee that is a graduate and a female?
Joint Probability

7/27
G
M
F
27/40
N
G
20/27
4/13
13/40
N
9/13

What is the
probability of
selecting an
employee that
is a female and
a graduate?
P(F and G) =
P(F)·P(G|F)
Complimentary Events

A box contains 20 DVDs, 4 of which are defective.
If two DVDs are selected at random (without
replacement) from this box, what is the
probability that both are defective?
D2

D1
G
D2
G
G
P(D1 and D2) =
P(D1)·P(D2|D1)
Probability of Independent Events

Is given the multiplication rule:
P(A and B) = P(A)·P(B)
Example 1:
 An office building has two fire detectors.
The probability is 0.03 that any fire
detector of this type will fail to go off
during a fire. Find the probability that
both fire detectors will fail to go off in
case of a a fire.
Probability of Independent Events

Is given the multiplication rule:
P(A and B) = P(A)·P(B)
Example 2:
 The probability that a patient is allergic
to penicillin is 0.20. Suppose this drug is
administered to 3 patients.


Find the probability that all 3 are allergic to it
Find the probability that at least ONE of them is not
allergic to it
Probability of Independent Events
0.2
B
C
0.2
C
0.8
0.2
B
A
Ā
0.8
0.2
B
0.8
B
0.8
C
0.2
C
0.8
C
0.2
C
0.8
C
0.2
C
0.8
Union of Events & The Addition Rule

Addition rule to find the probability of
union of events
P(A or B) = P(A)+P(B) – P(A and B)
A
Ā
B
Union of Events & The Addition Rule

A university president has proposed that
all students must take a course in ethics
as a requirement for graduation. 300
faculty members and students from this
university were asked about their
opinion on this issue.
Union of Events & The Addition Rule

What is the probability that a randomly
selected person from these 300 persons
is a faculty member or is in favor of this
proposal?
Favor
Oppose
Neutral
Total
Faculty
45
15
10
70
Student
90
110
30
230
Total
135
125
40
300