Chap 4- 1 Chapter 4 Basic Probability Business Statistics: A First Course 6 th

Learning Objectives
In this chapter, you learn:

Basic probability concepts
Conditional probability

Basic Probability Concepts
Probability – the chance that an uncertain event

Assessing Probability
There are three approaches to assessing the probability of

based on a combination of an individual’s past experience, personal opinion, and analysis of a particular situation

Assuming all outcomes are equally likely

probability of occurrence

probability of occurrence

Example of a priori probability
When randomly selecting a day from

the probability of selecting a face card (Jack, Queen, or

Example of empirical probability
Find the probability of selecting a male

Probability of male taking stats

Events
Each possible outcome of a variable is an event.

Simple event
An

Sample Space
The Sample Space is the collection of all

Visualizing Events
Contingency Tables -- For All Days in 2012



Decision Trees
All

Not Jan.

Jan.

Not Wed.

Wed.

Wed.

Not Wed.

Sample Space

Total
Number
Of
Sample
Space
Outcomes

4
27
48
287

Trees
Red 2

Black 2 24 26

Total 4 48 52

Ace Not Ace Total

Full Deck
of 52 Cards

Red Card

Black Card

Not an Ace

Ace

Ace

Not an Ace

Sample Space

Sample Space

2
24
2
24

= aces
Let B = red cards




A
B
A ∩ B = ace

A U B = ace or red

Definition: Simple Probability
Simple Probability refers to the probability of a

P(Jan.) = 31 / 366

P(Wed.) = 52 / 366

Not Wed. 27 287 314

Wed. 4 48 52

Total 31 335 366

Jan. Not Jan. Total

Definition: Joint Probability
Joint Probability refers to the probability of an

P(Jan. and Wed.) = 4 / 366

P(Not Jan. and Not Wed.)
= 287 / 365

Not Wed. 27 287 314

Wed. 4 48 52

Total 31 335 366

Jan. Not Jan. Total

Mutually Exclusive Events
Mutually exclusive events
Events that cannot occur simultaneously

Example: Randomly

one card from a deck of cards

A = queen

Collectively Exhaustive Events
Collectively exhaustive events
One of the events must occur

= aces; B = black cards;
C = diamonds; D

Computing Joint and Marginal Probabilities
The probability of a joint event,

Joint Probability Example
P(Jan. and Wed.)

Marginal Probability Example
P(Wed.)

P(A1 and B2)
P(A1)
Total
Event
Marginal & Joint Probabilities

P(A2 and B1)

P(A1 and B1)

Event

Total

1

Joint Probabilities

Marginal (Simple) Probabilities

A1

A2

B1

B2

P(B1)

P(B2)

P(A2 and B2)

P(A2)

Probability Summary So Far
Probability is the numerical measure of the

Certain

Impossible

0.5

1

0

0 ≤ P(A) ≤ 1 For any event A

If A, B, and C are mutually exclusive and collectively exhaustive

General Addition Rule
P(A or B) = P(A) + P(B) -

General Addition Rule:

If A and B are mutually exclusive, then
P(A and B) = 0, so the rule can be simplified:

P(A or B) = P(A) + P(B)
For mutually exclusive events A and B

General Addition Rule Example
P(Jan. or Wed.) = P(Jan.) + P(Wed.)

= 31/366 + 52/366 - 4/366 = 79/366

Don’t count the four Wednesdays in January twice!

2 black, 2 green, 1 yellow, and 0 blue marbles.

General Addition Rule Example

deck of 52 cards. Determine the probability of each event.

General Addition Rule Example

Computing Conditional Probabilities
A conditional probability is the probability of one

Where P(A and B) = joint probability of A and B
P(A) = marginal or simple probability of A
P(B) = marginal or simple probability of B

The conditional probability of A given that B has occurred

The conditional probability of B given that A has occurred

What is the probability that a car has a GPS

Conditional Probability Example

Of the cars on a used car lot, 90% have air conditioning (AC) and 40% have a GPS. 35% of the cars have both.

Conditional Probability Example
Of the cars on a used car lot,

(continued)

Conditional Probability Example
Given AC, we only consider the top row

(continued)

No GPS

GPS

Total

AC

0.35

0.55

0.90

No AC

0.05

0.05

0.10

Total

0.40

0.60

1.00

4-
What is the probability that a car has a CD

Conditional Probability Example

Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both.

Using Decision Trees
Has AC
Does not have AC
Has GPS
Does not have

Has GPS

Does not have GPS

P(AC)= 0.9

P(AC’)= 0.1

P(AC and GPS) = 0.35

P(AC and GPS’) = 0.55

P(AC’ and GPS’) = 0.05

P(AC’ and GPS) = 0.05

All
Cars

Given AC or no AC:

Conditional
Probabilities

Using Decision Trees
Has GPS
Does not have GPS
Has AC
Does not have

Has AC

Does not have AC

P(GPS)= 0.4

P(GPS’)= 0.6

P(GPS and AC) = 0.35

P(GPS and AC’) = 0.05

P(GPS’ and AC’) = 0.05

P(GPS’ and AC) = 0.55

All
Cars

Given GPS or no GPS:

(continued)

Conditional
Probabilities

Independence
Two events are independent if and only if:



Events A and

Multiplication Rules
Multiplication rule for two events A and B:
Note: If

and the multiplication rule simplifies to

3 red, 2 black, 2 green, 1 yellow, and 0

3 red, 2 black, 2 green, 1 yellow, and 0

Marginal Probability
Marginal probability for event A:





Where B1, B2, …, Bk

Bayes’ Theorem
Bayes’ Theorem is used to revise previously calculated probabilities

Bayes’ Theorem
where:
Bi = ith event of k mutually exclusive and

Bayes’ Theorem Example
A drilling company has estimated a 40% chance

Let S = successful well
U = unsuccessful

Bayes’ Theorem Example

(continued)

Bayes’ Theorem Example
(continued)
Apply Bayes’ Theorem:
So the revised probability of success,

Given the detailed test, the revised probability of a successful

Bayes’ Theorem Example

Sum = 0.36

(continued)

Counting Rules
Rules for counting the number of possible outcomes

Counting Rule

kn

Counting Rules
Counting Rule 2:
If there are k1 events on the

(k1)(k2)…(kn)

(continued)

Counting Rules
Counting Rule 3:
The number of ways that n items

n! = (n)(n – 1)…(1)

(continued)

Counting Rules
Counting Rule 4:
Permutations: The number of ways of arranging

(continued)

Counting Rules
Counting Rule 5:
Combinations: The number of ways of selecting

(continued)

Chapter Summary
Discussed basic probability concepts
Sample spaces and events, contingency tables,