Basic Probability Study Notes

CHAPTER 4: BASIC PROBABILITY

OBJECTIVES

After completing this chapter, you should be able to:
- Realize basic concepts about random experiments.
- Find the probability of events using the addition rules.
- Find the probability of compound events using the multiplication rule.
- Find the conditional probabilities of an event.

CONTENT OUTLINE

Sample Spaces and Probability
Addition Rules for Probability
Multiplication Rules & Conditional Probability
Permutations and Combinations

INTRODUCTION

Note: This PowerPoint is only a summary; the main source should be the book.

DEFINITION

Probability: The numerical measure of the likelihood that an event in the future will happen. It can also be defined as the chance of an event occurring.

SAMPLE SPACES AND PROBABILITY

A probability experiment is a chance process that leads to well-defined results called outcomes.
An outcome is the result of a single trial of a probability experiment.
A sample space is the set of all possible outcomes of a probability experiment, denoted by the symbol S.

SOME SAMPLE SPACES

Examples of sample spaces:
- Roll a die: S = {1, 2, 3, 4, 5, 6}
- Toss a coin: S = {Heads, Tails}
- Toss two coins: S = {HH, HT, TH, TT}
- Answer a true/false question: S = {True, False}

EVENTS

An event consists of outcomes of a probability experiment.
- Simple event: An event with one outcome.
- Compound event: An event containing more than one outcome.
- Example: Roll a die
  - Simple event: A = {6}
  - Compound events: B = Odd numbers = {1, 3, 5}, E = Even numbers = {2, 4, 6}.

TYPES OF PROBABILITY

Classical Probability
Empirical Probability (Relative Frequency)
Subjective Probability

CLASSICAL PROBABILITY

Definition: Classical probability uses sample spaces to determine the numerical probability that an event will happen and assumes that all outcomes in the sample space are equally likely to occur.
Equally Likely Events: Events that have the same probability of occurring.

Formula:

$P(E) = \frac{n(E)}{n(S)}$
- where n(E) = number of desired outcomes, and n(S) = total number of possible outcomes.

EXAMPLES

Example 1

Problem: Find the probability of obtaining a head and the probability of obtaining a tail for one toss of a coin.

Example 2

Problem: If a fair die is rolled one time, find the probability of:
1. A number 5
2. An even number
3. A number less than 3

CLASS WORK

When a single die is rolled, what is the probability of:
a) A number greater than 4
b) An odd number
c) A prime number
d) A number less than 7
e) A number 9.

Conditional Probability

Definition: Conditional Probability is the probability of an event occurring given that another event has occurred.

PROBABILITY RULES

For any event E: $0 \leq P(E) \leq 1$
If an event E cannot occur: $P(E) = 0$
If an event E is certain to occur: $P(S) = 1$
The sum of the probabilities of all the outcomes in the sample space is 1: $\sum P = 1$

COMPLEMENTARY EVENTS

Definition

The complement of event A, denoted by A' or A complement, is the event that includes all the outcomes for an experiment that are not in A.

Example

Find the complements of each event described:
- Event (E): Rolling a die and getting a 4 is complemented by getting a {1, 2, 3, 5, 6}.

Rule for Complementary Events

If the probability that a person lives in an industrialized country is $P = \frac{1}{4}$ , then the probability that a person does not live in an industrialized country is:
- $P(non-industrialized) = 1 - P = 1 - \frac{1}{4} = \frac{3}{4}$

EXAMPLES

In a study, 23% of people said vanilla was their favorite flavor of ice cream. Find the probability that the selected person's favorite flavor is not vanilla:
- $P(not\ vanilla) = 1 - P(vanilla) = 1 - 0.23 = 0.77 = 77\%$

CLASS WORK

If 65% of IT technicians in Hargeisa are male, find the probability that an IT technician in Hargeisa is female.

EMPIRICAL PROBABILITY

Relies on actual experience (frequency) to determine the likelihood of outcomes:

Formula

$P(E) = \frac{f}{n}$
- where f = frequency of desired class and n = sum of all frequencies.

Example

In a sample of 50 people:
- 21 had type O blood
- 22 had type A blood
- 5 had type B blood
- 2 had type AB blood
- Set up a frequency distribution and find the following probabilities:
  a) Probability of a person having type O blood.

CONDITIONAL PROBABILITY

Conditional probability formula:
$P(B|A) = \frac{P(A \cap B)}{P(A)}$
Examples of dependence: An urn contains 3 red balls, 2 blue balls, and 5 white balls. A ball is selected and noted, then replaced (not replaced). Find the probability of each of these scenarios.

CLASS WORK

Given the probabilities for events A and events B, find the following:
- P(A), P(B), P(A and B).

PERMUTATIONS AND COMBINATIONS

Definition of Permutation: An arrangement of objects in a specific order.
- Formula: $P(n,r) = \frac{n!}{(n-r)!}$
Definition of Combination: A grouping of objects where order does not matter.
- Formula: $C(n,r) = \frac{n!}{r!(n-r)!}$

EXAMPLES

Example of Permutation

A club has 20 members; select 3 office holders (president, secretary, treasurer). Find the total arrangements:
- $n = 20, r = 3$
- Total arrangements = $P(20,3) = \frac{20!}{(20-3)!} = 6840$

Example of Combination

Select 3 jurors from 5 individuals:
- Total combinations are given by: $C(5,3) = \frac{5!}{3!(5-3)!} = 10$

CLASS WORK

Tasks involving probabilities of different scenarios of drawing balls or arranging stories are provided for practice.

Examples with their solutions :

SAMPLE SPACES AND PROBABILITY

Example 1: Coin Toss

Problem: Find the probability of obtaining a head and a tail for one toss of a coin.
Solution:
- Sample Space $S = {Heads, Tails}$ , so $n(S) = 2$ .
- $P(Head) = \frac{1}{2} = 0.5$
- $P(Tail) = \frac{1}{2} = 0.5$

Example 2: Rolling a Die

Problem: If a fair die is rolled one time, find the probability of:
1. A number 5: $A = {5} \implies P(A) = \frac{1}{6}$
2. An even number: $B = {2, 4, 6} \implies P(B) = \frac{3}{6} = \frac{1}{2}$
3. A number less than 3: $C = {1, 2} \implies P(C) = \frac{2}{6} = \frac{1}{3}$

CLASS WORK: Die Rolling

When a single die is rolled, the probabilities are:
- a) A number greater than 4: Outcomes are ${5, 6}$ . $P = \frac{2}{6} = \frac{1}{3}$
- b) An odd number: Outcomes are ${1, 3, 5}$ . $P = \frac{3}{6} = \frac{1}{2}$
- c) A prime number: Outcomes are ${2, 3, 5}$ . $P = \frac{3}{6} = \frac{1}{2}$
- d) A number less than 7: Outcomes are ${1, 2, 3, 4, 5, 6}$ . $P = \frac{6}{6} = 1$ (Certain event)
- e) A number 9: No outcomes. $P = \frac{0}{6} = 0$ (Impossible event)

COMPLEMENTARY EVENTS

Rule and Examples

Rule: $P(A') = 1 - P(A)$
Vanilla Ice Cream: If 23% like vanilla, the probability they do not like vanilla is:
- $P(not\ vanilla) = 1 - 0.23 = 0.77 = 77\%$

CLASS WORK: IT Technicians

Problem: If 65% of IT technicians are male, find the probability that a selected technician is female.
Solution: $P(Female) = 1 - P(Male) = 1 - 0.65 = 0.35 = 35\%$

EMPIRICAL PROBABILITY

Example: Blood Types

Data: Total $n = 50$ . Type O = 21, Type A = 22, Type B = 5, Type AB = 2.
Problem: Find the probability of a person having type O blood.
Solution: $P(Type\ O) = \frac{f}{n} = \frac{21}{50} = 0.42 = 42\%$

CONDITIONAL PROBABILITY

Formula: $P(B|A) = \frac{P(A \cap B)}{P(A)}$
Example (Balls in Urn): 3 red, 2 blue, 5 white ( $Total = 10$ ).
- Probability of Red then Blue with replacement: $P(R) \times P(B) = \frac{3}{10} \times \frac{2}{10} = \frac{6}{100} = 0.06$
- Probability of Red then Blue without replacement: $P(R) \times P(B|R) = \frac{3}{10} \times \frac{2}{9} = \frac{6}{90} = \frac{1}{15} \approx 0.067$

PERMUTATIONS AND COMBINATIONS

Example of Permutation

Problem: Select 3 office holders (President, Secretary, Treasurer) from 20 members.
Solution: Order matters.
- P(20, 3) = 20! / (20 - 3)! = 20! / 17! = 6,840