Lifestyle journal of video presentation and mutual

1. Chapter 2
Probability Concepts
and Applications

2. Objectives
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

3. Probability
Life is uncertain and full of surprise. Do you
know what happen tomorrow
Make decision and live with the consequence
The probability of an event is a numerical value
that measures the likelihood that the event can
occur

4. 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 ( ) 1
P A
 

5. 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.

6. 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
  

7. P(A or B)
+ -
=
P(A) P(B) P(A and B)
P(A or B)

8. 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)

9. 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

10. 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)

11. 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


12. 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?

13. Probability(A|B)
/
P(A|B) = P(AB)/P(B)
P(AB) P(B)
P(A)

14. 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

15. 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
0  R  100
Test the
lifetime of a
light bulb
(minutes)
Time bulb
lasts - up to
80,000
minutes
S = time bulb
burns
0  S  80,000

16. Probability Distributions
Table 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 2.5
Probability
Function

17. Expected Value of a Discrete Probability
Distribution



n
i
i
i )
X
(
P
X
)
X
(
E
1
2.9
)
1
.
0
)(
1
(
)
3
.
0
)(
2
(
)
3
.
0
)(
3
(
)
2
.
0
)(
4
(
)
1
.
0
)(
5
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
5
5
4
4
3
3
2
2
1
1
5
1











 

X
P
X
X
P
X
X
P
X
X
P
X
X
P
X
X
P
X
X
E i
i
i

18. Variance of a Discrete Probability
Distribution
 
   
i
n
i
i X
P
X
E
X




1
2
2

       
   
29
.
1
0.361
0.243
0.003
0.242
-
0.44
)
1
.
0
(
)
9
.
2
1
(
)
3
.
0
(
2.9)
-
(2
3
.
0
9
.
2
3
2
.
0
9
.
2
4
1
.
0
9
.
2
5
2
2
2
2
2
2
















19. Binomial Distribution
Assumptions:
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

20. Binomial Distribution
r
n
r
q
p 

r)!
-
(n
r!
n!
Probability of r successes
in n trials
n = number of trials
r = number of successes
p = probability of success
q = probability of failure

21. Binomial Distribution
)
p
(
np
np



1




22. 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

23. 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
e
X
f

24. Normal Distribution for Different Values of 
0
30 40 50 60 70
=50 =60
=40

25. 0 0.5 1 1.5 2
Normal Distribution for Different Values of

=0.1
=0.2
=0.3
 = 1

26. Three Common Areas
Under the Curve
Three Normal distributions with different
areas

27. Three Common Areas
Under the Curve
Three Normal
distributions
with different
areas

28. The Relationship Between
Z and X
55 70 85 100 115 130 145
-3 -2 -1 0 1 2 3




x
Z
=100
=15

29. Haynes Construction Company
Example
Fig. 2.12

30. Haynes Construction Company
Example
Fig. 2.13

31. Haynes Construction Company
Example
Fig. 2.14

32. The Negative Exponential
Distribution
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1 1.2
x
e
)
X
(
f 
 

=5
Expected value = 1/
Variance = 1/2

33. 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 
 
