Chapter 2 Probability Concepts and Applications. Objectives Students will be able to: – Understand...

Chapter 2Chapter 2

Probability Concepts Probability Concepts and Applicationsand Applications

ObjectivesObjectives

Students will be able to:

– Understand the basic foundations of probability

analysis

– Do basic statistical analysis

– Know various type of probability distributions

and know when to use them

Probability

Life is uncertain and full of surprise. Do you know what happen tomorrowMake decision and live with the consequenceThe probability of an event is a numerical value that measures the likelihood that the event can occur

Basic Probability Properties

Let P(A) be the probability of the event A, then The sum of the probability of all possible outcomes

should be 1.

0 ( ) 1P A

Mutually Exclusive Events

Two events are mutually exclusive if they can not occur at the same time. Which are mutually exclusive?

• Draw an Ace and draw a heart from a standard deck of 52 cards

• It is raining and I show up for class• Dr. Li is an easy teacher and I fail the class• Dr. Beaubouef is a hard teacher and I ace the class.

Addition Rule of Probability

If two events A and B are mutually exclusive, then

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

( or ) ( ) ( ) ( and )P A B P A P B P A B

P(A or B)

+ -

=

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

P(A or B)

Independent and Dependent

Events are either– statistically independent (the occurrence of one

event has no effect on the probability of

occurrence of the other) or

– statistically dependent (the occurrence of one

event gives information about the occurrence of

the other)

Which Are Independent?

(a) Your education

(b) Your income level

(a) Draw a Jack of Hearts from a full 52 card deck

(b) Draw a Jack of Clubs from a full 52 card deck

(a) Chicago Cubs win the National League pennant

(b) Chicago Cubs win the World Series

Conditional Probability

Conditional probability

the probability of event B given that event A

has occurred P(B|A) or, the probability of

event A given that event B has occurred P(A|

B)

Multiplication Rule of Probability

If two events A and B are mutually exclusive, Otherwise,

( and ) ( ) ( | ) ( ) ( | )P A B P A P B A P B P A B

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

Joint Probabilities, Dependent Events

Your stockbroker informs you that if the stock market

reaches the 10,500 point level by January, there is a

70% probability the Tubeless Electronics will go up in

value. Your own feeling is that there is only a 40%

chance of the market reaching 10,500 by January.

What is the probability that both the stock market will

reach 10,500 points, and the price of Tubeless will go

up in value?

Probability(A|B)

/

P(A|B) = P(AB)/P(B)

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

Random Variables

Discrete random variable - can assume only a finite

or limited set of values- i.e., the number of

automobiles sold in a year

Continuous random variable - can assume any one

of an infinite set of values - i.e., temperature,

product lifetime

Random Variables (Numeric)Experiment Outcome Random Variable Range of

Random Variable

Stock 50 Xmas trees

Number of trees sold

X = number of trees sold

0,1,2,, 50

Inspect 600 items

Number acceptable

Y = number acceptable

0,1,2,…, 600

Send out 5,000 sales letters

Number of people e responding

Z = number of people responding

0,1,2,…, 5,000

Build an apartment building

%completed after 4 months

R = %completed after 4 months

0R100

Test the lifetime of a light bulb (minutes)

Time bulb lasts - up to 80,000 minutes

S = time bulb burns

0S80,000

Probability DistributionsTable 2.4

Outcome X Number Responding

P(X)

SA 5 10 0.10

A 4 20 0.20

N 3 30 0.30

D 2 30 0.30

SD 1 10 0.10

D

0.00

0.05

0.10

0.15

0.20

0.25

0.30

1 2 3 4 5

Figure 2.5Probability Function

Expected Value of a Discrete Probability Distribution

n

iii )X(PX)X(E

2.9

)1.0)(1()3.0)(2(

)3.0)(3()2.0)(4()1.0)(5(

)()(

)()()(

)()(

5544

332211

5

1

XPXXPX

XPXXPXXPX

XPXXE ii

i

Variance of a Discrete Probability Distribution

i

n

ii XPXEX

1

22

29.1

0.3610.2430.0030.242- 0.44

)1.0()9.21(

)3.0(2.9)-(2 3.09.23

2.09.241.09.25

2

22

222

Binomial DistributionAssumptions:

1. Trials follow Bernoulli process – two possible outcomes

2. Probabilities stay the same from one trial to the next

3. Trials are statistically independent

4. Number of trials is a positive integer

Binomial Distribution

rnr qp r)!-(nr!

n!

Probability of r successes

in n trials

n = number of trials

r = number of successes

p = probability of success

q = probability of failure

Binomial Distribution

)p(np

np

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

1 2 3 4 5 6

(r) Number of Successes

P(r)

N = 5, p = 0.50

Binomial Distribution

Probability Distribution Continuous Random Variable

Probability density function - f(X)

5 5.05 5.1 5.15 5.2 5.25 5.3 5.35 5.4

Normal Distribution

2

2)(2/1

2

1)(

X

eXf

Normal Distribution for Different Values of

0

30 40 50 60 70

=50 =60=40

0 0.5 1 1.5 2

Normal Distribution for Different Values of

=0.1

=0.2=0.3

= 1

Three Common Areas Under the Curve

Three Normal distributions with different areas

Three Common Areas Under the Curve

Three Normal

distributions

with different

areas

The Relationship Between Z and X

55 70 85 100 115 130 145

-3 -2 -1 0 1 2 3

x

Z

=100

=15

Haynes Construction Company Example Fig. 2.12

Haynes Construction Company ExampleFig. 2.13

Haynes Construction Company ExampleFig. 2.14

The Negative Exponential Distribution

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1 1.2

xe)X(f

=5

Expected value = 1/Variance = 1/2

The Poisson Distribution

0.00

0.05

0.10

0.15

0.20

0.25

0.30

1 2 3 4 5 6 7 8 9

=2

Expected value = Variance = !X

e)X(P

x