Chap 4-1

Business Statistics: A Decision-Making Approach

7th Edition

Chapter 4Using Probability and

Probability Distributions

4-2

Important Terms Random Variable - Represents a possible numerical value

from a random event and it can vary from trial to trial Probability – the chance that an uncertain event will occur Experiment – a process that produces outcomes for

uncertain events Sample Space (or event) – the collection of all possible

experimental outcomes

4-3

Basic Rule of Probability

Individual Values Sum of All Values

0 ≤ P(Ei) ≤ 1

For any event Ei

1)P(ek

1ii

where:k = Number of individual outcomes in the sample spaceei = ith individual outcome

The value of a probability is between 0 and 1

0 = no chance of occurring1 = 100% change of occurring

The sum of the probabilities of all the outcomes in a sample space must = 1 or 100%

Simple probability

The probability of an event is the number of favorable outcomes divided by the total number of possible equally-likely outcomes. This assumes the outcomes are all equally weighted. Textbook: classical probability

Chap 4-4

Chap 4-5

Visualizing Events (or sample space)

Contingency Tables

Tree Diagrams

Red 2 24 26 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

AceNot an Ace

Sample Space

Sample Space2

24

2

24

Chap 4-6

Experimental Outcome Example A automobile consultant records fuel type and vehicle

type for a sample of vehicles 2 Fuel types: Gasoline, Diesel 3 Vehicle types: Truck, Car, SUV

6 possible experimental outcomes: e1 Gasoline, Truck e2 Gasoline, Car e3 Gasoline, SUV e4 Diesel, Truck e5 Diesel, Car e6 Diesel, SUV

Gasoline

Diesel

CarTruck

Truck

Car

SUV

SUV

e1

e2

e3

e4

e5

e6

Simple probability

What is the probability that a card drawn at random from a deck of cards will be an ace? 52 cards in the deck and 4 are aces The probability is ???? Each card represents a possible outcome 52

possible outcomes.

Chap 4-7

Simple probability

Chap 4-8

There are 36 possible outcomes when a pair of dice is thrown.

Simple probability

Calculate the probability that the sum of the two dice will be equal to 5? Four of the outcomes have a total of 5: (1,4; 2,3;

3,2; 4,1) Probability of the two dice adding up to 5 is ????.

Chap 4-9

Simple probability

Calculate the probability that the sum of the two dice will be equal to 12? Only one (6,6) ????

Chap 4-10

Simple Probability

The probability of event A is denoted by P(A) Example: Suppose a coin is flipped 3 times. What is the

probability of getting two tails and one head? The sample space consists of 8 possible outcomes. S = {TTT, TTH, THT, THH, HTT, HTH, HHT, HHH} the probability of getting any particular outcome is 1/8. Getting two tails and one head: A = {TTH, THT, HTT} P(A) = ????

Chap 4-11

Essential Concepts and Rules of Probability

Independent event Dependent event Joint and Conditional Probabilities Mutually Exclusive Event Addition Rule Complement Rule Multiplication Rule

Chap 4-12

Chap 4-13

Independent: Occurrence of one does not influence the

probability of occurrence of the other

Dependent: Occurrence of one affects the probability of the

other

Independent vs. Dependent Events

Chap 4-14

Independent EventsA = heads on one flip of fair coinB = heads on second flip of same coin

Result of second flip does not depend on the result of the first flip.

Dependent EventsX = rain forecasted on the newsY = take umbrella to work

Probability of the second event is affected by the occurrence of the first event

Examples

Probability Types

Simple (Marginal) probability Involves only a single random variable, the outcome of which is

uncertain Joint probability

Involves two or more random variables, in which the outcome of all is uncertain

Conditional probability (will be discussed soon!) Download “Probability Type Example ” Excel file

Chap 4-15

Exercise 4-22-A (page 158)

Chap 4-16

Download “Computer Use” Answer for questions using “Pivot Table”

Exercise 4-22-B (page 158)

Chap 4-17

Practice

Exercise 4-23: try questions.

Chap 4-18

  Electrical Mechanical Total

Lincoln 28 39 67

Tyler 64 69 133

Total 92 108 200

Conditional Probability: ex 1 A conditional probability is the probability of an event

given that another event has occurred. Involves two or more random variables, in which the outcome of

at least one is known Conditional probability for any two events A , B:

Chap 4-19

P(B))andP(AB)|P(A B

0P(B)where

Notation

Chap 4-20

What is the probability that a car has a CD player, given that it has AC (use the table on the next slide)?

i.e., we want to find P(CD | AC)

Conditional Probability: ex 1

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.

Chap 4-21

Conditional Probability: ex 1

No CDCD TotalAC .2 .5 .7No AC .2 .1 .3Total .4 .6 1.0

What is the probability that a car has a CD player, given that it has AC ?

.2857.7.2

P(AC)AC)andP(CDAC)|P(CD

Conditional Probability: ex 2

What is the probability that the total of two dice will be greater than 8 given that the first die is a 6? This can be computed by considering only outcomes

for which the first die is a 6. Then, determine the proportion of these outcomes that total more than 8.

Chap 4-22

Conditional Probability: ex 2

There are 6 outcomes for which the first die is a 6.

And of these, there are four that total more than 8.

(6,3; 6,4; 6,5; 6,6)

Chap 4-23

Conditional Probability: ex 2

6 outcomes for which the first die is a 6 There are four that total more than 8 The probability of a total greater than 8 given that the

first die is 6 is 4/6 = 2/3. More formally, this probability can be written as:

p(total>8 | Die 1 = 6) = 2/3(6,3; 6,4; 6,5; 6,6). Conditional probability using “Probability Type

Example” Excel file

Chap 4-24

Chap 4-25

Mutually Exclusive Events

If A occurs, then B cannot occur A and B have no common elements

Black Cards

Red Cards

A card cannot be Black and Red at the same time.

AB

Mutually Exclusive Events

If events A and B are mutually exclusive, then the probability of A or B is p(A or B) = p(A) + p(B)

What is the probability of rolling a die and getting either a 1 or a 6?

impossible to get both a 1 and a 6 p(1 or 6) = p(1) + p(6) = 1/6 + 1/6 = 1/3

Chap 4-26

Chap 4-27

Not Mutually Exclusive Events

P(A or B) = P(A)+ P(B) – P(A and B)

A B

P(A or B) = P(A) + P(B) - P(A and B)Overlap: joint probability

■ If the events are not mutually exclusive,

A B+ =

Not Mutually Exclusive Events

What is the probability that a card will be either an ace or a spade?

p(ace) = 4/52 and p(spade) = 13/52 The only way both can be drawn is to draw the ace of

spades. There is only one ace of spades: p(ace and spade) = 1/52 .

The probability of an ace or a spade can be: p(ace) + p(spade) - p(ace and spade) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13

Chap 4-28

Rule of Addition

Suppose we have two events and want to know the probability that either event occurs.

Mutually exclusive: P(A or B) = P(A) + P(B)

Not Mutually exclusive: P(A or B) = P(A)+ P(B) – P(A and B)

Chap 4-29

Chap 4-30

Complement Rule

The complement of an event E is the collection of all possible elementary events not contained in event E. The complement of event E is represented by E.

Complement Rule:

)EP(1P(E)

E

E1)EP(P(E)

Or,

If there is a 40% ( E ) chance it will rain,

there is a 60% ( E ) chance it won’t rain!

Rule of Multiplication

The rule of multiplication applies to the situation when we want to know the probability that both events (event A and event B) occur. Addition: either events

Independent: P(A ∩ B) = P(A) P(B) Dependent: P(A ∩ B) = P(A) P(B|A)

Chap 4-31

Independent Events

If A and B are independent, then the probability that events A and B both occur is: p(A and B) = p(A) x p(B)

What is the probability that a coin will come up with heads twice in a row?

Two events must occur: a head on the first toss and a head on the second toss.

The probability of each event is 1/2 the probability of both events is: 1/2 x 1/2 = 1/4.

Chap 4-32

Independent Events

What is the probability that the first card is the ace of clubs (put it back: replace) and the second card is a club (any club)? The probability of the first event is 1/52 (only one ace

of club) The probability of the second event is 13/52 = 1/4

(composed of clubs) Answer is 1/52 x 1/4 = 1/208

Chap 4-33

Dependent Events

If A and B are not independent, then the probability of A and B both occur is:

p(A and B) = p(A) x p(B|A) where p(B|A) is the conditional probability of B given A.

Chap 4-34

Dependent Events

If someone draws a card at random from a deck and then, without replacing the first card, draws a second card, what is the probability that both cards will be aces? 

4 of the 52 cards are aces First: p(A) = 4/52 = 1/13.

Of the 51 remaining cards, 3 are aces. So, p(B|A) = 3/51 = 1/17

Probability of A and B is: 1/13 x 1/17 = 1/221.

Chap 4-35

Practice

Page 180 Exercise 4-26 Exercise 4-30

Chap 4-36

Summary

Probability Rules Addition Rule for Two Events Addition Rule for Mutually Exclusive Events Conditional Probability Multiplication Rules

Chap 4-37