This document provides an introduction to basic probability theory concepts. It defines key terms like random experiment, sample space, sample point, event, mutually exclusive events, and independent events. Examples are given to illustrate these concepts, such as listing the sample spaces and events for experiments like rolling dice, tossing coins, and selecting students. The properties of the impossible event (empty set) and sure event (sample space) are also discussed. Overall, the document aims to explain the basic foundations of probability theory and provide practice applying these concepts to solve probability problems.

1. Statistics and Probability
PROBABILITY THEORY (PART 1)
1st Semester SY 2021-2022
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2. 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
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3. Statistics and Probability: Probability Theory
“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
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4. Statistics and Probability: Probability Theory
“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}
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5. Statistics and Probability: Probability Theory
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.
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6. Statistics and Probability: Probability Theory
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 }
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7. Statistics and Probability: Probability Theory
“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.
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8. Statistics and Probability: Probability Theory
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
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9. Statistics and Probability: Probability Theory
Consider the random experiment of rolling a die.
The sample space is given to be:
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}
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10. Statistics and Probability: Probability Theory
Consider the experiment of tossing a pair of dice.
The sample space is 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) }
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
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11. Statistics and Probability: Probability Theory
Consider the experiment of tossing a pair of dice.
The sample space is 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) }
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) }
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12. Statistics and Probability: Probability Theory
Consider the experiment of tossing a pair of dice.
The sample space is 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) }
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) }
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13. Statistics and Probability: Probability Theory
“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.
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14. Statistics and Probability: Probability Theory
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15. Statistics and Probability: Probability Theory
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16. Statistics and Probability: Probability Theory
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17. Statistics and Probability: Probability Theory
Consider the experiment of tossing a pair of dice.
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
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18. Statistics and Probability: Probability Theory
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) }
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) }
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19. Statistics and Probability: Probability Theory
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) }
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) }
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20. Statistics and Probability: Probability Theory
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) }
A ∩ B ∩ C = intersection of A, B, C
= 𝜙
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21. Statistics and Probability: Probability Theory
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 }
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22. Statistics and Probability: Probability Theory
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!!!
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23. Statistics and Probability: Probability Theory
“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
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24. Statistics and Probability: Probability Theory
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25. Statistics and Probability: Probability Theory
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
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26. Statistics and Probability: Probability Theory
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.
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27. Statistics and Probability: Probability Theory
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.
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28. Statistics and Probability: Probability Theory
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.
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29. Statistics and Probability: Probability Theory
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.
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30. Statistics and Probability: Probability Theory
e. Region 3, 6, 7 and 8
together
𝑴𝑪
The family will not have a
mechanical problem.
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31. Statistics and Probability: Probability Theory
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.
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32. Statistics and Probability: Probability Theory
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
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33. Statistics and Probability: Probability Theory
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
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34. Statistics and Probability: Probability Theory
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
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35. Statistics and Probability: Probability Theory
d. The family will not
arrive at a beach hotel
with no vacancies.
4, 5, 7, 8
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36. Statistics and Probability
PROBABILITY THEORY (PART 1)
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