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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