SAMPLE SPACES and PROBABILITY (3).pptx

CHAPTER 4: PROBABILITY AND COUNTING
RULES

Probability
It is the study of randomness and uncertainty.
The chance of an event occurring.
In the early days, probability was associated
with games of chance (gambling).
Examples: card games, slot machines,
lotteries, insurance, investments, weather
forecasting and in other various areas.
It is also the basis of inferential statistics.

Basic Concepts of Probability
 Probability Experiment
A chance process that leads to well-
defined results called outcomes.
Example:
Flipping a coin
Rolling a die
Drawing a card from a deck

 Outcome
The result of a single trial of a
probability experiment.
 Trial
Means flipping a coin once, rolling
one die once, or the like.

 Sample Space
The set of all possible outcomes of a
Experiment Sample space
Toss a coin Head, Tail
Roll a die 1,2,3,4,5,6,
Answer a true/false
question
True, False
Toss two coins Head-Head, Tail-Tail,
Head-Tail, Tail-Head

Tree Diagram
It is a device consisting of line
segments emanating from a starting
point and also from the outcome point.
It is used to determine all possible
outcomes of a probability experiment.
Event
It consists of a set of outcomes of a

Ex. Use a tree diagram to find the sample space
for the gender of three children in a family

Two types of Event
• Simple Event
An event with one outcome (rolling a
die one time, choosing one card)
• Compound Event
An event with more than one outcome
(rolling an odd number on one die -3
possibilities)

Three basic interpretations
of probability:
1. Classical probability
2. Empirical or relative frequency
probability
3. Subjective probability

Classical Probability
Uses sample spaces to determine numerical
probability that an event will happen.
 An experiment is not performed to
determine the probability of an event.
 Assumes that all outcomes in a sample
space are equally likely to occur (6
possibilities on a die have equally likely
chance of occurring)
Equally likely events are events that have
the same probability of occurring.

Example:
When a single die is rolled, each outcome has
the same probability of occurring. Since there
are 6 outcomes, each outcome has a
probability of
𝟏
𝟔
.
Example:
When a card is selected from an ordinary deck
of 52 cards, you assume that the deck has
been shuffled, and each card has the same
probability of being selected. Each outcome
has the possibility of
𝟏
𝟓𝟐
.
Classical Probability

The probability of any event E is
Number of outcomes in E
Total number of outcomes in the Sample Space
The probability is denoted by
𝑃 𝐸 =
𝑛(𝐸)
𝑛(𝑆)
Answers given as fractions, decimals or
percentages
Formula for Classical
Probability

Rounding rules for
Probabilities
 Reduced fractions or decimals rounded to
two or three decimal places
 If probability is extremely small, round the
decimal to the first nonzero digit after the
decimal point.
(0.000000478 = 0.0000005)

Example: Drawing Cards
Find the probability of getting a red ace when
a card is drawn at random from an ordinary
deck of cards.
Solution:
• Since there are 52 cards and there are 2 red aces,
namely, the ace of hearts and the ace of diamonds,
P(red ace) =
𝟐
𝟓𝟐
=
𝟐
𝟐𝟔.
Rounding rules for
Probabilities

Probability Rules
The probability of any event E is a number
(either a fraction or a decimal) between and
including 0 and 1. This is denoted by 0 ≤ P(E) ≤
1.
- It state that probabilities cannot be
negative or greater than 1.
Probability Rules
Probability Rule 1

Probability Rules
If an event E cannot occur (the event contains
no members in the sample space), its probability
is 0.
Example: Rolling a Die
When a single die is rolled, find the probability of
getting a 9.
• Solution: Since the sample space is 1, 2, 3, 4, 5,
and 6, it is impossible to get a 9. Hence, the
probability is P(9) = 0.
Probability Rules
Probability Rule 2

Probability Rules
If an event E is certain, then the probability of
E is 1.
Example: Rolling a Die
When a single die is rolled, what is the
probability of getting a number less than 7?
Solution
Since all outcomes—1, 2, 3, 4, 5, and 6—are less than 7,
the probability is P(number less than 7) =
6
6
= 1
The event of getting a number less than 7 is certain.
Probability Rules
Probability Rule 3

Probability Rules
The sum of the probabilities of all the outcomes
in the sample space is 1.
Example: The roll of a fair die, each outcome in
the sample space has a probability of
𝟏
𝟔
.
Probability Rules
Probability Rule 4

Complementary Events
 It is another important concept in
probability theory.
 The Complement of event E is the set of
outcomes in the sample space that are not
included in the outcomes of event E. The
complement of E is denoted by Ē (E “Bar”).

Rules for Complementary
Events
• P(Ē) = 1- P(E) or P(E) = 1- P(Ē) or P(E) +
P(Ē) = 1
• Stated in words, the rule is: If the
probability of an event or the probability of
its complement is known, then the other
can be found by subtracting the probability
from 1.

Venn Diagrams
• Used to pictorially represent the probability
of events.
• Venn Diagram for the probability and
complement:
P(E) P(E)
P(S) = 1 P(Ē)

Empirical Probability
 The type of probability that uses
frequency distributions based on
observations to determine numerical
probabilities of events.
 It relies on actual experience to
determine the likelihood of outcomes.

Formula for Empirical
Probability
Given a frequency distribution, the probability of
an event being in a given class is
𝑷 𝑬 =
𝒇𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝒇𝒐𝒓 𝒕𝒉𝒆 𝒄𝒍𝒂𝒔𝒔
𝒕𝒐𝒕𝒂𝒍 𝒇𝒓𝒆𝒒𝒖𝒆𝒎𝒄𝒊𝒆𝒔 𝒊𝒏 𝒕𝒉𝒆 𝒅𝒊𝒔𝒕𝒓𝒊𝒃𝒖𝒕𝒊𝒐𝒏
=
𝒇
𝒏
This probability is called empirical probability
and is based on observation.

Law of Large Numbers
 When a probability experiment is
repeated a large number of times, the
relative frequency probability of an
outcome will approach its theoretical
probability.

Subjective Probability
 The type of probability that uses a
probability value based on an educated
guess or estimate, employing opinions
and inexact information.
 In subjective probability, a person or
group makes an educated guess at the
chance that an event will occur. This
guess is based on the person’s
experience and evaluation of a solution.

PROBABILITY AND
RISK - TAKING
 An area in which people fail to
understand probability is risk taking.
 Honestly, people fear situations or
events that have a relatively small
probability of happening rather than
those events that have a greater
likelihood of occurring.

Group 1:
• MIRALLES, VICTOR A.
• CUBA, JOHN ALMER D.
• ALTICEN, PAMELA JEAN C.
• KATANGKATANG, JANINE
• CABANGISAN, CHRISTINE MAE M.
• LOCSIN, JESSA MAE

SAMPLE SPACES and PROBABILITY (3).pptx

SAMPLE SPACES and PROBABILITY (3).pptx

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