1. MAT 446 Supplementary Note for Ch 2
Keller: Stats for Mgmt & Econ, 7th Ed
July 31, 2018
MAT 446 Supplementary Note for Ch 2
Myung Song, Ph.D.
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

2. Sample Spaces and Events
1) S = Sample Space: This is a set, or list, of all possible outcomes of a random process (experiment).
2) An event is a subset of the sample space.
Ex) Coin Toss:
S = {Head, Tail}
An event can be ∅ or{Head} or {Tail} or {Head, Tail}

3. Sample Spaces It’s the question that determines the sample space.
H HHH
M … M
M HHM
H HMH
M HMM …
S = { HHH, HHM, HMH, HMM, MHH, MHM, MMH, MMM }
Note: 8 elements, 23
A. A basketball player shoots three free throws. What are the possible sequences of hits (H) and misses (M)?
B. A basketball player shoots three free throws. What is the number of baskets made?
S = { 0, 1, 2, 3 }

4. Some Relations from Set Theory
There are several types of combinations and relationships between events:
Complement event
Intersection of events
Union of events
Disjoint (Mutually Exclusive) events

5. A Ac Complement of an Event
The complement of event A is defined to be the event consisting of all sample points that are “not in A”.
Complement of A is denoted by Ac
The Venn diagram below illustrates the concept of a complement. A Ac

6. A Ac Complement of an Event
For example, the rectangle stores all the possible tosses of 2 dice {(1,1), 1,2),… (6,6)} Let A = tosses totaling 7 {(1,6),
(2, 5), (3,4), (4,3), (5,2), (6,1)} A Ac

7. Intersection of Two Events
The intersection of events A and B is the set of all sample points that are in both A and B.
The intersection is denoted: A and B A B

8. Intersection of Two Events
For example, let A = tosses where first toss is 1 {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)}
and B = tosses where the second toss is 5 {(1,5), (2,5), (3,5), (4,5), (5,5), (6,5)}
The intersection is {(1,5)} A B

9. A B Union of Two Events (6.1)
The union of two events A and B, is the event containing all sample points that are in A or B or both:
Union of A and B is denoted: A or B A B

10. Union of Two Events
For example, let A = tosses where first toss is 1 {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)}
and B is the tosses that the second toss is 5 {(1,5), (2,5), (3,5), (4,5), (5,5), (6,5)}
Union of A and B is {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)}
(2,5), (3,5), (4,5), (5,5), (6,5)} A B

11. Disjoint (Mutually Exclusive) Events
When two events are disjoint or mutually exclusive (that is the two events cannot occur together), their joint probability is 0, hence: A B
Mutually exclusive; no points in common…
For example A = tosses totaling 7 and B = tosses totaling 11

12. Basic Relationships of Events (6.1)
Complement of Event
Union of Events A Ac A B
Intersection of Events
Mutually Exclusive Events A B A B

13. Interpreting Probability
There are some ways to assign a probability:
Objective Interpretation (Relative Frequency) : Proportion of outcomes observed in the long run that comprised the event (based on experimentation or historical data.)
Subjective Interpretation : Assigning probabilities based on the assignor’s subjective judgment.

14. Relative Frequency
Bits & Bytes Computer Shop tracks the number of desktop computer systems it sells over a month (30 days):
For example,
10 days out of 30
2 desktops were sold.
From this we can construct
the probabilities of an event
(i.e. the # of desktop sold on a given day)…
Desktops Sold
# of Days 1 2 10 3 12 4 5

15. Relative Frequency
Desktops Sold
# of Days 1
1/30 = .03 2
2/30 = .07 10
10/30 = .33 3 12
12/30 = .40 4 5
5/30 = .17
∑ = 1.00
“There is a 40% chance Bits & Bytes will sell 3 desktops on any given day”

16. Limiting Relative Frequency
If a situation, trial, or experiment is repeated again and again, the proportion of success will tend to to approach the probability that any one outcome will be a success.
ex) the proportion of heads in a "large" number of coin flips "should be" roughly 1/2

17. Limiting Relative Frequency
The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials.
First series of tosses
Second series

18. Subjective Interpretation
“In the subjective interpretation, we define probability as the degree of belief that we hold in the occurrence of an event”
E.g. weather forecasting’s “P.O.P.”
“Probability of Precipitation” (or P.O.P.) is defined in different ways by different forecasters, but basically it’s a subjective probability based on past observations combined with current weather conditions.
POP 60% – based on current conditions, there is a 60% chance of rain (say).

19. Conditional Probability
Conditional probabilities reflect how the probability of an event can change if we know that some other event has occurred/is occurring.
Example: The probability that a cloudy day will result in rain is different if you live in Los Angeles than if you live in Seattle.
Our brains effortlessly calculate conditional probabilities, updating our “degree of belief” with each new piece of evidence.

20. Conditional Probability
Conditional probability is used to determine how two events are related; that is, we can determine the probability of one event given the occurrence of another related event.
Conditional probabilities are written as P(A | B) and read as “the probability of A given B” and is calculated as:

21. Conditional Probability
Again, the probability of an event given that another event has occurred is called a conditional probability…
Note how “A given B” and “B given A” are related…

22. Conditional Probability
The Multiplication Rule for P(A and B) :

23. Independence
One of the objectives of calculating conditional probability is to determine whether two events are related.
In particular, we would like to know whether they are independent, that is, if the probability of one event is not affected by the occurrence of the other event.
Two events A and B are said to be independent if
P(A|B) = P(A) or
P(B|A) = P(B)

24. Independence P(A and B) when events are independent:
A and B are independent events, iff

25. Example: Conditional Probability
Q: The probability that a bus from Philadelphia to New York will leave on time(5:00pm) is 0.8, and the probability that it will leave on time and also arrive on time(10:00pm) is 0.6.
What is the probability that a bus that leaves on time will also arrive on time?
(b) If the probability that such a bus will arrive on time is 0.7, what is the probability that a bus that arrives on time also left on time?

26. Example: Multiplication Rule
Q: Suppose that the probability is 0.4 that a skin cancer is diagnosed correctly and the probability is 0.6 that the patient will be cured if diagnosed correctly. What is the probability that a person who has the cancer will be diagnosed correctly and cured?

27. Example: Multiplication Rule
Q: A panel of jurors consists of 10 persons who have had high school education and 6 persons who have had college education. If two persons are randomly selected , what is the probability that both of them will have had college education?

28. Example: Multiplication Rule (Independence)
Q: Check the following pair of events A and B, whether they are independent (a) If P(A)=0.2, P(B)=0.3, and P(A' and B)=0.24 (b) If P(A)=0.4, P(B')=0.3, and P(A and B)=0.29

29. Example: Multiplication Rule (General and Special)
Q: If two cards are drawn at random from 52 playing cards, what is the probabilities that they will both be diamonds if
The first card is replaced before the second card is drawn.
(with replacement)
(b) The first card is not replaced before the second card is drawn.
(without replacement)