This document serves as a guide for students to understand the fundamentals of probability, including the definitions of sample spaces, events, and types of probability approaches. It provides examples of random experiments, operations on events, and various probability rules, such as the complement, additive, and conditional rules. The document also includes practical applications and exercises to help students apply their knowledge to real-world situations.

2. Objectives:
1. At the end of this chapter, the students are expected to:
2. List down the elements of sample spaces and events;
3. Perform operations on events;
4. Demonstrate knowledge and understanding of the
probability of events and outcomes; and,
5. Use probability to analyze and understand real world
situations and problems.
2

3. RANDOM EXPERIMENTS is a process or procedure, repeatable under
basically the same condition, leading to well-defined outcomes. It is
random because we can never tell in advance what the realization is going
to be even if we can specify the possible outcomes.
SAMPLE SPACE is the set of all possible outcomes of a random experiment.
It is denoted by Greek letter omega (Ω) or S. It is also known as universal
set.
OUTCOME is the result of the experiment.
EVENTS is a subset of the sample space denoted by any letter in the English
alphabet. An event is an outcome of a random experiment. 3

4. 1. Given the random experiment: Tossing a coin and a die at
the same time. Give 3 examples of possible outcomes,
identify the sample space and give one possible event.
Solution:
Possible Outcomes: H1, T2, H3 (answers may vary)
Sample Space: {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5,
T6}
Possible event: Let A – be the event of obtaining a head and
an odd number (answers may vary)
4

5. 2. List the sample space of the experiment asking three students about
having or not having a facebook account.
Solution:
Let Y – be the representation of a student response having a facebook
account
N – be the representation of a student response not having a facebook
account
Since there are three students ask, each outcome must be composed of a
combination of three responses, these are as follows:
S = {YYY, YYN, YNN, NYY, NNY, YNY, NYN, NNN} 5

6. 3. From the experiment given on number 2, let B – event of
obtaining a response that is exactly 2 students have their
facebook account. List down the outcomes on event B.
Solution:
Since the condition given is that exactly 2 students have
their facebook account, one must look in the sample space
and select the outcomes which contain 2 “Y”.
Therefore, B = {YYN, NYY, YNY}
6

7. PROBABILITY is a number
between 0 and 1 inclusive,
associated with the
likelihood of occurrence
of a given event.
7

8. Types of Probability
1. Subjective approach
- It is a personal assessment of the likelihood of occurrence of an
event based on all evidence available.
Examples:
- A doctor may use subjective probability to assess that a certain
person with a rare disease will survive the operation.
- A teacher may ask each student what is their chances of passing the
subject and get different responses.
8

9. 2. Relative frequency or empirical approach
- It is the proportion of times that a particular outcome occurs in a very large
number of observations or experiments. It is the relative frequency of an event
in past occurrences.
Examples:
- If for every ten people who enter a department store on a payday, three make
at least one purchase then we can estimate the chance that a person entering
this department store on a payday will make at least one purchase is 3/10.
- Some field representative of the Environmental Protection Agency are doing
spot checks of water pollution in streams. Historically, 8 out of 10 such tests
produce favorable results, that is, no pollution.
9

10. 3. Classical or priori approach
- If an experiment can result in N equally likely outcomes and an event
A can result in n outcomes, then the probability of event A can be
obtained as:
10

11. a. What is the probability of getting an odd number in rolling
a die?
b. A box contains18 balls where 3 are red, 8 are blue, 6 are
yellow, and the rest are green. Find the probability of
getting a red ball.
c. A class consists of 25 girls and 15 boys. If a student is
selected from the class what is the probability of selecting
a girl?
11

12. 1. The Complement Rule
2. The Additive Rule
3. The Multiplicative Rule
4. The Conditional Rule
12

13. 13
Examples:
1. The probability that a student will get a passing grade in Probability and
Statistics under Sir Hernan is 0.63. What is the probability that the student
will not get a passing grade?
2. A certain SHS class is composed of 42 students. If there are 24 males,
what percentage is composed of females?

14. 1. For non-mutually exclusive events A and B, the 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 +
𝑃 𝐵 − 𝑃(𝐴 ∩ 𝐵).
2. For mutually exclusive events A and B, the 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 .
3. If 𝐴1,𝐴2, 𝐴3, … . . , 𝐴𝑛 are mutually exclusive, then
4. 𝑃 𝐴1 ∪ 𝐴2 ∪ 𝐴3 ∪ ⋯ . .∪ 𝐴𝑛 = 𝑃 𝐴1 + 𝑃 𝐴2 + 𝑃 𝐴3 + ⋯ + 𝑃 𝐴𝑛 =
𝑃 Ω = 1
14

15. Examples
A. The probability that a student will pass Filipino is 50% and the probability that he will pass
English is 80%. The probability that he will pass both is 60%. What is the probability that
the student will pass at least one of the two subjects?
B. In a college class attending a lecture hall, 36 are enrolled in Algebra, 40 are enrolled in
Filipino, and 24 students are enrolled in both subjects. If the total is 52 students, what is
the probability that a is enrolled in Algebra only?
C. A recent survey of banks revealed the following distribution for the interest rate being
charged on a home loan (based on a 30-year mortgage with a 10% down payment, as seen
in the table below). If a bank is selected at random from this distribution, what is the
chance that the interest rate charged on a home loan will exceed 8.0%?
15
Interest Rate 7.0% 7.5% 8.0% 8.5% more than 8.5%
Probability 0.11 0.22 0.24 0.35 0.08

16.  For dependent events A and B, the 𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 ∙ 𝑃 𝐵|𝐴
 For independent events A and B, the 𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 ∙ 𝑃 𝐵
 If 𝐴1,𝐴2, 𝐴3, … . . , 𝐴𝑛 are independent events, then
𝑃 𝐴1 ∩ 𝐴2 ∩ 𝐴3 ∩ ⋯ . .∩ 𝐴𝑛 = 𝑃 𝐴1 ∙ 𝑃 𝐴2 ∙ 𝑃 𝐴3 ∙ ⋯ ∙ 𝑃 𝐴𝑛
Note: Two events are independent if the occurrence or nonoccurrence
of one has no effect on the probability of occurrence of the other,
otherwise dependent.
16

17. a. The probability that Ian passes a stat exam is 0.79. The
probability Marc passes a stat exam, on the other hand, is
0.77. If they will not help each other out in a major exam,
what is the probability that both will pass?
b. A bag of fruits contains six mangoes, four atis and five
guavas. What is the probability of getting a mango and an
atis in that order if you are sampling
b1. without replacement? b2. with replacement?
17

18. The probability of an event B occurring when it is known
that some event A has occurred is called a conditional
probability. It is defined by:
𝑃 𝐵 𝐴 =
𝑃(𝐴∩𝐵)
𝑃(𝐴)
𝑖𝑓 𝑃(𝐴) > 0.
𝑃 𝐵 𝐴 − read as probability of B given A
18

19. a. The probability that a randomly selected student from college is a
freshman is 0.25 and the joint probability that the student is a math
major and a freshman is 0.05. Find the probability that a student
selected at random is a math major given that he/she is a freshman.
b. Two coins were tossed in succession. What is the probability that the
second toss would show a head if it is known that the first toss showed
tail?
c. A card was pulled out from a regular deck of cards. What is the
probability that it is
a. A red card if it is known to be a face card?
b. A face card if it is known to be a red card?
19

20. c. DNHS-SHS has a population 1,046 enrollees. They are divided
into two tracks, namely, Academic and TVL. The Academic Track is
composed of 327 students, of which 204 are females. The TVL
Track, on the other hand, is comprised by 425 males. If a student
is collected randomly, what is the probability that the student will
be
1. A male?
2. Coming from the TVL track?
3. A female given that she is enrolled in the Academic Track?
4. An Academic Track student which is known to be a female?
5. A male and from the TVL Track? 20

21. 21
From the collection of
shapes, what is the
probability of
randomly picking
1. A happy face?
2. A sad face?
3. A triangle or a
heart?
4. A triangle and a
moon?