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.

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

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
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,

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
0,1,2,, 50
Inspect 600
items
Number
acceptable
Y = number
0,1,2,…,
600
Send out
5,000 sales
letters
people e
responding
Z = number of
people responding
5,000
Build an
apartment
building
%completed
after 4
months
R = %completed
after 4 months R
100
Test the
lifetime of a
light bulb
(minutes)
Time bulb
lasts -
up to
80,000
minutes
S = time bulb
burns S

16. Probability Distributions
Figure 2.5
Probability Function

17. Expected Value of a Discrete Probability Distributionå = n i ) X ( P E 1

18. Variance of a Discrete Probability Distribution

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
n = number of trials
r = number of successes
p = probability of success
q = probability of failure
Probability of r successes
in n trials

21. Binomial Distribution

22. Binomial Distribution
N = 5, p = 0.50

23. Probability Distribution Continuous Random Variable
Normal Distribution
Probability density function - f(X)

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

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

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
=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
Expected value = 1/
Variance = 1/2
=5

33. The Poisson Distribution
Expected value = 
Variance = 
=2