Lecture-1-Probability-Theory-Part-1.pdf

Statistics and Probability
PROBABILITY THEORY (PART 1)
1st Semester SY 2021-2022
1

Statistics and Probability: Probability Theory
❖Define the terms related to basic probability theory
❖Illustrate the properties of a probability function
❖Define mutually exclusive events, conditional
probability and independent events
❖Apply basic concepts in probability theory
❖Solve problems involving probabilities
2

“A random experiment is a process that can be
repeated under similar conditions but whose outcome
cannot be predicted with certainty beforehand.”
Examples:
→Tossing a pair of dice
→Tossing a coin
→Selecting 5 cards from a well-shuffled deck of cards
→Selecting a sample of size n from a population of N using a
probability sampling method
3

“The sample space, denoted by Ω (omega), is the
collection of all possible outcomes of a random
experiment. An element of the sample space is called a
sample point.”
Examples:
→Rolling a die:
Ω = {1,2,3,4,5,6}
→Tossing a coin:
Ω = {head, tail}
4

Examples:
List the elements of the sample spaces of the following random
experiments:
1. Tossing a coin twice
2. Tossing a pair of dice
3. Four students are selected at random from a senior high
school and classified as grade 11 or grade 12 student. Using
the letter J for grade 11 and S for grade 12, list the elements
of the sample space.
5

1. Tossing a coin twice
Ω = { HH, HT, TH, TT }
2. Tossing a pair of dice
Ω = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5),
(2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5),
(4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5),
(6,6) }
3. Four students are selected at random from a senior high school and
classified as grade 11 or grade 12 student. Using the letter J for grade 11 and
S for grade 12, list the elements of the sample space.
Ω = { JJJJ, SSSS, JSSS, JJSS, JJJS, SSSJ, SSJJ, SJJJ, SJSJ, JSJS, SSJS, JJSJ, SJSS, JSJJ,
JSSJ, SJJS }
6

“An event is a subset of the sample space whose
probability is defined. We say that an event occurred if
the outcome of a random experiment is one of the
elements belonging in the event. Otherwise, the event did
not occur.”
We will use any capital Latin letter to denote an event of interest.
7

Consider the random experiment of rolling a die.
The sample space is given to be:
Ω = {1,2,3,4,5,6}
List the elements of the following events…
1. A = event of observing odd number of dots in a roll of a die
2. B = event of observing even number of dots in a roll of a die
3. C = event of observing less than 3 dots in a roll of a die
8

Consider the random experiment of rolling a die.
Ω = {1,2,3,4,5,6}
A = event of observing odd number of dots in a roll of a die
= {1,3,5}
B = event of observing even number of dots in a roll of a die
= {2,4,6}
C = event of observing less than 3 dots in a roll of a die
= {1,2}
9

Consider the experiment of tossing a pair of dice.
Ω = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1),
(3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3),
(5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) }
List the elements of the following events:
1. A = event of having the same number of dots on both dice
2. B = event of 3 dots on one die
3. C = event of getting a sum of 5 dots on both dice
4. D = event of 7 dots on one die
5. E = event of not having the same number of dots on both
dice
10

Ω = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1),
(3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3),
(5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) }
A = event of having the same number of dots on both dice
= { (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) }
B = event of 3 dots on one die
= { (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (1,3), (2,3), (4,3),
(5,3), (6,3) }
C = event of getting a sum of 5 dots on both dice
= { (1,4), (2,3), (3,2), (4,1) }
11

Ω = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1),
(3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3),
(5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) }
D = event of 7 dots on one die
= 𝜙
E = event of not having the same number of dots on both dice
= { (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,3), (2,4), (2,5),
(2,6), (3,1), (3,2), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,5),
(4,6), (5,1), (5,2), (5,3), (5,4), (5,6), (6,1), (6,2), (6,3), (6,4),
(6,5) }
12

“The impossible event is the empty set 𝟇”
“The sure event is the sample space Ω.”
Note: These two subsets of the sample space will always be events.
Remember that an event occurs if the outcome of the experiment belongs in
it. But 𝜙 is the empty set so it does not contain any elements and thus, it is
impossible for this event to happen.
On the other hand, Ω is the sample space so it contains all possible outcomes
of the experiment and thus we are sure that it will always occur.
13

14

15

16

The sample space is again given to be:
Ω = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6),
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1),
(5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) }
From “Event: Example 2”, we know the following:
A = { (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) }
B = { (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (1,3), (2,3), (4,3), (5,3), (6,3) }
C = { (1,4), (2,3), (3,2), (4,1) }
Find the following:
1. 𝐴𝐶 2. A ∪ B 3. A ∩ B 4. A ∪ B ∪ C 5. A ∩ B ∩ C
17

A = { (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) }
B = { (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (1,3), (2,3), (4,3), (5,3), (6,3) }
C = { (1,4), (2,3), (3,2), (4,1) }
𝑨𝑪
= sample points in the sample space but not in A
= { (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2),
(3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,6),
(6,1), (6,2), (6,3), (6,4), (6,5) }
A ∪ B = union of A and B
= { (1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (3,1), (3,2), (3,4), (3,5), (3,6), (1,3),
(2,3), (4,3), (5,3), (6,3) }
18

A = { (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) }
B = { (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (1,3), (2,3), (4,3), (5,3), (6,3) }
C = { (1,4), (2,3), (3,2), (4,1) }
A ∩ B = intersection of A and B
= { (3,3) }
A ∪ B ∪ C = union of A, B and C
= { (1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (3,1), (3,2), (3,4), (3,5),
(3,6), (1,3), (2,3), (4,3), (5,3), (6,3) (1,4), (4,1) }
19

A = { (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) }
B = { (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (1,3), (2,3), (4,3), (5,3), (6,3) }
C = { (1,4), (2,3), (3,2), (4,1) }
A ∩ B ∩ C = intersection of A, B, C
= 𝜙
20

2. An engineering firm is hired to determine if certain
waterways in Visayas are safe for fishing. Samples are taken
from three rivers. Getting the sample space for the three rivers
where F means safe for fishing and N is not safe for fishing:
Ω = { FFF, NFF, FFN, NFN, NNN, FNF, FNN, NNF } – 8 sample pts.
Your task is to define an event Y in words that has the
following elements:
Y = { FFF, NFF, FFN, NFN }
21

SOLUTION:
Define an event Y in words that has the following sample points:
Y = { FFF, NFF, FFN, NFN }
To answer this question, we must ask this…
WHAT IS THE SIMILARITIES BETWEEN THE ELEMENTS OF Y???
Note that each element of Y has “F” as the characteristic of the
SECOND RIVER. “F” means that IT IS SAFE FOR FISHING. Hence,
Answer:
Y is the EVENT that the SECOND RIVER IS SAFE FOR FISHING!!!
22

“Two events A and B are mutually exclusive if and only
if A ∩ B = 𝜙, that is, A and B have no elements in
common.”
Example:
➢ A and 𝐴𝐶
➢ A ∩ B and A ∩ 𝐵𝐶
➢ A and B ∩ 𝐴𝐶
➢ A and 𝜙
23

24

Suppose that a family decided to leave Manila to spend their vacation in a
beach town. Let M be the event that they will experience mechanical
problems, T is the event that they will receive a ticket for committing a
traffic violation, and V is the event that they will arrive at a beach hotel
with no vacancies.
1. Referring to the Venn diagram below, state in words the events
represented by the following regions:
a. Region 5
b. Region 3
c. Region 1 and 2 together
d. Region 4 and 7 together
e. Region 3, 6, 7, and 8 together
25

a. Region 5
𝑻𝑪 ∩ 𝑽𝑪 ∩ 𝑴
The family will not
receive a ticket and will
not arrive at a beach
hotel with no vacancies
but will have a
mechanical problem.
26

b. Region 3
𝑻 ∩ 𝑽 ∩ 𝑴𝑪
The family will receive a
ticket and will arrive at a
beach hotel with no
vacancies but will not
have a mechanical
problem.
27

c. Region 1 and 2 together
𝑴 ∩ 𝑽
The family will have a
mechanical problem and
will arrive at a beach
hotel with no vacancies.
28

d. Region 4 and 7 together
𝑽𝑪 ∩ 𝑻
The family will not arrive
at a beach hotel with no
vacancies and will
receive a ticket for
committing a traffic
violation.
29

e. Region 3, 6, 7 and 8
together
𝑴𝑪
The family will not have a
mechanical problem.
30

2. Referring to the same Venn diagram, list the
numbers of the regions represented by the following
events:
a. The family will experience no mechanical
problem and will not receive a ticket for
violation but will arrive at a beach hotel with
no vacancies.
b. The family will experience both mechanical
problems and trouble in locating a hotel with
vacancy but will not receive a ticket for traffic
violation.
c. The family will either have mechanical trouble
or arrive at a beach hotel with no vacancies
but will not receive a ticket for traffic violation.
d. The family will not arrive at a beach hotel with
no vacancies.
31

a. The family will
experience no
mechanical problem
and will not receive a
ticket for violation
but will arrive at a
beach hotel with no
vacancies.
6
32

b. The family will
experience both
mechanical problems
and trouble in locating a
hotel with vacancy but
will not receive a ticket
for traffic violation.
2
33

c. The family will either
have mechanical trouble
or arrive at a beach hotel
with no vacancies but
will not receive a ticket
for traffic violation.
2, 5, 6
34

d. The family will not
arrive at a beach hotel
with no vacancies.
4, 5, 7, 8
35

Statistics and Probability
PROBABILITY THEORY (PART 1)
36

Lecture-1-Probability-Theory-Part-1.pdf

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