Probability

Chap 2-1

Concepts & Applications

Basic Probability

chap 2-2

Important Terms

Random Experiment – a process leading to an uncertain outcome

A coin is thrown

A consumer is asked which of two products he

or she prefers

The daily change in an index of stock market

prices is observed

chap 2-3

3

Sample Space - Collection of all possible outcomes

e.g.: All six faces of a die:

e.g.: All 52 cardsa deck of bridge cards

Important Terms

chap 2-4

Event – any subset of basic outcomes from the sample space

Important Terms

Simple Event

Outcome With 1 Characteristic

Red card from a deck of bridge cards

Ace card from a deck of bridge cards

Joint Event

2 Events Occurring Simultaneously

A and B, (AB):

Red, ace card from a bridge deck

Male, over age 20

Important Terms

chap 2-6

Compound Event

One or Another Event Occurring

D or E, (DE): Ace or Red card from bridge deck

Important Terms

chap 2-7

Important Terms

Intersection of Events – If A and B are two events in a sample space S, then the intersection, A ∩ B, is the set of all outcomes in S that belong to both A and B

(continued)

A BAB

S

chap 2-8

Important Terms

A and B are Mutually Exclusive Events if they have no basic outcomes in common i.e., the set A ∩ B is empty

(continued)

A B

S

chap 2-9

Important Terms

Union of Events – If A and B are two events in a sample space S, then the union, A U B, is the set of all outcomes in S that belong to either

A or B

(continued)

A B

The entire shaded area represents A U B

S

chap 2-10

Important Terms

Events E1, E2, … Ek are Collectively Exhaustive events if E1 U E2 U . . . U Ek = S i.e., the events completely cover the sample space

The Complement of an event A is the set of all basic outcomes in the sample space that do not belong to A. The complement is denoted

(continued)

A

AS

A

chap 2-11

Examples

Let the Sample Space be the collection of all possible outcomes of rolling one die:

S = [1, 2, 3, 4, 5, 6]

Let A be the event “Number rolled is even”

Let B be the event “Number rolled is at least 4”

Then

A = [2, 4, 6] and B = [4, 5, 6]

chap 2-12

(continued)

Examples

S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6]

5] 3, [1, A

6] [4, BA

6] 5, 4, [2, BA

S 6] 5, 4, 3, 2, [1, AA

Complements:

Intersections:

Unions:

[5] BA

3] 2, [1, B

chap 2-13

Mutually exclusive: A and B are not mutually exclusive

The outcomes 4 and 6 are common to both

Collectively exhaustive: A and B are not collectively exhaustive

A U B does not contain 1 or 3

(continued)

Examples

S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] B = [4, 5, 6]

chap 2-14

Probability

Probability – the chance that an uncertain event will occur (always between 0 and 1)

0 ≤ P(A) ≤ 1 For any event A

Certain

Impossible

.5

1

0

chap 2-15

Assessing Probability

There are three approaches to assess the probability of an uncertain event:

1. classical probability

Assumes all outcomes in the sample space are equally likely to

occur

spacesampletheinoutcomesofnumbertotal

eventthesatisfythatoutcomesofnumber

N

NAeventofyprobabilit A

chap 2-16

Assessing Probability

Three approaches (continued)

2. relative frequency probability

the limit of the proportion of times that an event A occurs in a large number of trials, n

3. subjective probability

an individual opinion or belief about the probability of occurrence

populationtheineventsofnumbertotal

Aeventsatisfythatpopulationtheineventsofnumber

n

nAeventofyprobabilit A

chap 2-17

Probability Postulates

1. If A is any event in the sample space S, then

2. Let A be an event in S, and let Oi denote the basic outcomes. Then

(the notation means that the summation is over all the basic outcomes in A)

3. P(S) = 1

1P(A)0

)P(OP(A)A

i

chap 2-18

Probability Rules

The Complement rule:

The Addition rule: The probability of the union of two events is

1)AP(P(A)i.e., P(A)1)AP(

B)P(AP(B)P(A)B)P(A

chap 2-19

Addition Rule Example

Consider a standard deck of 52 cards, with four suits:

♥ ♣ ♦ ♠

Let event A = card is an Ace

Let event B = card is from a red suit

Find P(A or B)?

chap 2-20

P(Red U Ace) = P(Red) + P(Ace) - P(Red ∩ Ace)

= 26/52 + 4/52 - 2/52 = 28/52Don’t count the two red aces twice!

BlackColor

Type Red Total

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

(continued)

Addition Rule Example

chap 2-21

Counting the Possible Outcomes

Use the Combinations formula to determine the number of combinations of n things taken k at a time

where n! = n(n-1)(n-2)…(1) 0! = 1 by definition

k)!(nk!

n! Cn

k

chap 2-22

Example

The sample space contains 5 A’s and 7 B’s. What is the probability that a randomly selected set of 2 will include 1 A and 1 B?

Sol:

53.066

35

66

75)1&1(

)1&1(122

71

51

BAP

C

CCBAP

chap 2-23

Practice Questions

Q # 1: The sample space contains 6 red and 4 green marbles. What is the probability that a randomly selected set of 3 will include 1 Red & 2 Green marbles.

Ans: 36/120Q # 2: ABC Inc. is hiring managers to till four key positions. The candidates are five men and three women. Assuming that every combination of men and women is equally likely to be chosen, what is the probability that at least one woman will be selected?

Ans: 13/14Hint: A: at least one woman is selected

Use complement law

chap 2-24

A conditional probability is the probability of one event, given that another event has occurred:

P(B)

B)P(AB)|P(A

P(A)

B)P(AA)|P(B

The conditional probability of A given that B has occurred

The conditional probability of B given that A has occurred

Conditional Probability

chap 2-25

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

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

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.

chap 2-26

Conditional Probability Example

No CDCD Total

AC .2 .5 .7

No AC .2 .1 .3

Total .4 .6 1.0

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.

.2857.7

.2

P(AC)

AC)P(CDAC)|P(CD

(continued)

chap 2-27

Conditional Probability Example

No CDCD Total

AC .2 .5 .7

No AC .2 .1 .3

Total .4 .6 1.0

Given AC, we only consider the top row (70% of the cars). Of these, 20% have a CD player. 20% of 70% is 28.57%.

.2857.7

.2

P(AC)

AC)P(CDAC)|P(CD

(continued)

chap 2-28

Multiplication Rule

Multiplication rule for two events A and B:

also

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

P(A)A)|P(BB)P(A

Question

chap 2-29

If a card is selected, at random, from a deck of 52

cards, what is the probability that the card is an ace

of red color?

chap 2-30

Multiplication Rule Example

P(Red ∩ Ace) = P(Red| Ace)P(Ace)

BlackColor

Type Red Total

Ace 2 2 4

Non-Ace 24 24 48

Total 26 26 52

52

2

52

4

4

2

52

2

cards of number total

ace and red are that cards of number

chap 2-31

Statistical Independence

Two events are statistically independent if and only if:

Events A and B are independent when the probability of one event is not affected by the other event

If A and B are independent, then

P(A)B)|P(A

P(B)P(A)B)P(A

P(B)A)|P(B

if P(B)>0

if P(A)>0

chap 2-32

Statistical Independence Example

No CDCD Total

AC .2 .5 .7

No AC .2 .1 .3

Total .4 .6 1.0

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.

Are the events AC and CD statistically independent?

• Out of a target audience of 2,000,000, ad Out of a target audience of 2,000,000, ad A A reaches reaches 500,000 viewers, 500,000 viewers, BB reaches 300,000 viewers and both reaches 300,000 viewers and both ads reach 100,000 viewers.ads reach 100,000 viewers.

• What is What is PP((AA | | BB)?)?

500,000( ) .25

2,000,000P A

300,000( ) .15

2,000,000P B

100,000( ) .05

2,000,000P A B

Statistical IndependenceStatistical IndependenceStatistical IndependenceStatistical Independence

Example: Television AdsExample: Television Ads

( ) .05( | ) .30

( ) .15

P A BP A B

P B

.3333 or 33%.3333 or 33%

• So, So, PP(ad (ad AA) = .25) = .25 PP(ad (ad BB) = .15) = .15 PP((AA BB) = .05) = .05 PP((AA | | BB) = .3333 ) = .3333

• PP((AA | | BB) = .3333 ≠ ) = .3333 ≠ PP((AA) = .25) = .25

• PP((AA))PP((BB)=(.25)(.15)=.0375 ≠ )=(.25)(.15)=.0375 ≠ PP((AA BB)=.05 )=.05

• Are events Are events AA and and BB independent? independent?

Independent EventsIndependent EventsIndependent EventsIndependent Events

Example: Television AdsExample: Television Ads

chap 2-35

Q:1 A single 6-sided die is rolled. What is the probability of each outcome? What is the probability of rolling an even number? of rolling an odd number?Q:2 A single 6-sided die is rolled. What is the probability of rolling a 2 or a 5?Q:3 A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single marble is chosen at random from the jar, what is the probability of choosing a red marble? a green marble? a blue marble? a yellow marble?Q:4 A glass jar contains 1 red, 3 green, 2 blue, and 4 yellow marbles. If a single marble is chosen at random from the jar, what is the probability that it is yellow or green?

Practice Questions

chap 2-36

Q:5 On New Year's Eve, the probability of a person having a car accident is 0.09. The probability of a person driving while intoxicated is 0.32 and probability of a person having a car accident while intoxicated is 0.15. What is the probability of a person driving while intoxicated or having a car accident?Q:6 A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing a card that is not a king?Q:7 A card is chosen at random from a deck of 52 cards. It is then replaced and a second card is chosen. What is the probability of choosing a jack and an eight?

Practice Questions

Q # 8: A hamburger chain found that 75% of all customers use mustard, 80% use ketchup, and 65% use both.

a.What is the probability that a customer will use at least one of these? (0.90)

b. What is the probability that a ketchup user uses mustard? (0.8125)

Q # 9: The probability of A is 0.70 and the probability of B is 0.80 and the probability of both is 0.50. What is the conditional probability of A given B? Are A and B statistically independent?

Practice Questions

chap 2-38

Practice Questions

Q # 10: (a) A fair die is rolled twice. Let A be the event of an odd total and B the event of an ace on the first die. Verify that P(A/B) = P(A) (b) The probability that a man will be alive in 25 years is 3/5 and that his wife will be alive in 25 years is 2/3. Find the probability that (i) both will be alive (ii) At least one will be alive.

Q # 11: A music store owner finds that 25% of the customers entering the store ask as assistant for help and 30% of the customers make a purchase before leaving. It is also found that 20% of all customers both ask for assistance and make a purchase. What is the probability that a customer does at least one of these two things?

Q # 12: The accompanying table shows proportions of adults categorized to whether they are readers or non-readers of newspapers and whether or not they voted in the last election.

Practice Questions

Voted Readers Non-readers

Yes 0.63 0.13

No 0.14 0.10

What is the probability that a randomly chosen adult from this population who did not read newspaper did not vote? (0.4348)

chap 2-40

Q # 13: In a certain college, 4% of the men and 1% of the women are taller than 6 feet. Furthermore 60% of the students are women. Now, if a student is selected at random and is taller than 6 feet, what is the probability that the student is a woman?

Practice Questions