1. Random Experiments Probability Rules of Probability Independent Events5
Probability (Part 1)
Chapter
Random Experiments
Probability
Rules of Probability
Independent Events
McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

2. Random Experiments Sample Space
A random experiment is an observational process whose results cannot be known in advance.
The set of all outcomes (S) is the sample space for the experiment.
A sample space with a countable number of outcomes is discrete.

3. Random Experiments Sample Space
For example, when Citibank makes a consumer loan, the sample space is:
S = {default, no default}
The sample space describing a Wal-Mart customer’s payment method is:
S = {cash, debit card, credit card, check}

4. Random Experiments Sample Space
For a single roll of a die, the sample space is:
S = {1, 2, 3, 4, 5, 6}
When two dice are rolled, the sample space is the following pairs:
{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
S =
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5. It would be impractical to enumerate this sample space.
Random Experiments
Sample Space
Consider the sample space to describe a randomly chosen United Airlines employee by: 2 genders, 21 job classifications, 6 home bases (major hubs) and 4 education levels
There are: 2 x 21 x 6 x 4 = 1008 possible outcomes
It would be impractical to enumerate this sample space.
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6. Random Experiments Sample Space S = {all X such that X > 0}
If the outcome is a continuous measurement, the sample space can be described by a rule.
For example, the sample space for the length of a randomly chosen cell phone call would be
S = {all X such that X > 0}
or written as S = {X | X > 0}
The sample space to describe a randomly chosen student’s GPA would be
S = {X | 0.00 < X < 4.00}

7. Random Experiments Events
An event is any subset of outcomes in the sample space.
A simple event or elementary event, is a single outcome.
A discrete sample space S consists of all the simple events (Ei):
S = {E1, E2, …, En}

8. Are these two events equally likely to occur?
Random Experiments
Consider the random experiment of tossing a balanced coin. What is the sample space?
Events
S = {H, T}
What are the chances of observing a H or T?
These two elementary events are equally likely.
When you buy a lottery ticket, the sample space S = {win, lose} has only two events.
Are these two events equally likely to occur?

9. Random Experiments Events (Figure 5.1)
A compound event consists of two or more simple events.
For example, in a sample space of 6 simple events, we could define the compound events
A = {Music, DVD, VH}
B = {Newspapers,
Magazines}
These are displayed in a Venn diagram:

10. Random Experiments Events
Many different compound events could be defined.
Compound events can be described by a rule.
For example, the compound event A = “rolling a seven” on a roll of two dice consists of 6 simple events:
S = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}

11. Probability Definitions
The probability of an event is a number that measures the relative likelihood that the event will occur.
The probability of event A [denoted P(A)], must lie within the interval from 0 to 1:
0 < P(A) < 1
If P(A) = 0, then the event cannot occur.
If P(A) = 1, then the event is certain to occur.

12. Probability Definitions
In a discrete sample space, the probabilities of all simple events must sum to unity:
P(S) = P(E1) + P(E2) + … + P(En) = 1
For example, if the following number of purchases were made by
credit card:
32%
debit card:
20%
cash:
35%
check:
18%
Sum =
100%
P(credit card) =
.32
P(debit card) =
.20
P(cash) =
.35
P(check) =
.18
Sum =
1.0
Probability

13. Probability
Businesses want to be able to quantify the uncertainty of future events.
For example, what are the chances that next month’s revenue will exceed last year’s average?
How can we increase the chance of positive future events and decrease the chance of negative future events?
The study of probability helps us understand and quantify the uncertainty surrounding the future.

14. Probability

15. Probability What is Probability? Three approaches to probability:
Example
Empirical
There is a 2 percent chance of twins in a randomly- chosen birth.
Classical
There is a 50 % probability of heads on a coin flip.
Subjective
There is a 75 % chance that England will adopt the Euro currency by 2010.

16. Probability Empirical Approach
Use the empirical or relative frequency approach to assign probabilities by counting the frequency (fi) of observed outcomes defined on the experimental sample space.
For example, to estimate the default rate on student loans:
P(a student defaults) = f /n
number of defaults
number of loans =

17. Probability Empirical Approach
Necessary when there is no prior knowledge of events.
As the number of observations (n) increases or the number of times the experiment is performed, the estimate will become more accurate.

18. Probability Law of Large Numbers
The law of large numbers is an important probability theorem that states that a large sample is preferred to a small one.
Flip a coin 50 times. We would expect the proportion of heads to be near .50.
However, in a small finite sample, any ratio can be obtained (e.g., 1/3, 7/13, 10/22, 28/50, etc.).
A large n may be needed to get close to .50.
Consider the results of 10, 20, 50, and 500 coin flips.

19. Probability
(Figure 5.2)

20. Probability Practical Issues for Actuaries
Actuarial science is a high-paying career that involves estimating empirical probabilities.
For example, actuaries - calculate payout rates on life insurance, pension plans, and health care plans - create tables that guide IRA withdrawal rates for individuals from age 70 to 99

21. Probability Practical Issues for Actuaries
Challenges that actuaries face:
- Is n “large enough” to say that f/n has become a good approximation to P(A)?
- Was the experiment repeated identically?
- Is the underlying process invariant over time?
- Do non-statistical factors override data collection?
- What if repeated trials are impossible?

22. Probability Classical Approach
In this approach, we envision the entire sample space as a collection of equally likely outcomes.
Instead of performing the experiment, we can use deduction to determine P(A).
a priori refers to the process of assigning probabilities before the event is observed.
a priori probabilities are based on logic, not experience.

23. Probability Classical Approach
For example, the two dice experiment has 36 equally likely simple events. The P(7) is
The probability is obtained a priori using the classical approach as shown in this Venn diagram for 2 dice:

24. Probability Subjective Approach
A subjective probability reflects someone’s personal belief about the likelihood of an event.
Used when there is no repeatable random experiment.
For example, - What is the probability that a new truck product program will show a return on investment of at least 10 percent? - What is the probability that the price of GM stock will rise within the next 30 days?

25. Probability Subjective Approach
These probabilities rely on personal judgment or expert opinion.
Judgment is based on experience with similar events and knowledge of the underlying causal processes.

26. Rules of Probability Complement of an Event
The complement of an event A is denoted by A′ and consists of everything in the sample space S except event A.

27. Rules of Probability Complement of an Event
Since A and A′ together comprise the entire sample space, P(A) + P(A′ ) = 1
The probability of A′ is found by P(A′ ) = 1 – P(A)
For example, The Wall Street Journal reports that about 33% of all new small businesses fail within the first 2 years. The probability that a new small business will survive is:
P(survival) = 1 – P(failure) = 1 – .33 = .67 or 67%

28. Rules of Probability Odds of an Event
The odds in favor of event A occurring is
The odds against event A occurring is

29. Rules of Probability Odds of an Event
Odds are used in sports and games of chance.
For a pair of fair dice, P(7) = 6/36 (or 1/6). What are the odds in favor of rolling a 7?

30. Rules of Probability Odds of an Event
On the average, for every time a 7 is rolled, there will be 5 times that it is not rolled.
In other words, the odds are 1 to 5 in favor of rolling a 7.
The odds are 5 to 1 against rolling a 7.
In horse racing and other sports, odds are usually quoted against winning.

31. Rules of Probability Odds of an Event
If the odds against event A are quoted as b to a, then the implied probability of event A is:
P(A) =
For example, if a race horse has a 4 to 1 odds against winning, the P(win) is
P(win) =
or 20%

32. Rules of Probability Union of Two Events
(Figure 5.5)
The union of two events consists of all outcomes in the sample space S that are contained either in event A or in event B or both (denoted A  B or “A or B”).
 may be read as “or” since one or the other or both events may occur.

33. Rules of Probability Union of Two Events
For example, randomly choose a card from a deck of 52 playing cards.
If Q is the event that we draw a queen and R is the event that we draw a red card, what is Q  R?
It is the possibility of drawing either a queen (4 ways) or a red card (26 ways) or both (2 ways).

34. Rules of Probability Intersection of Two Events
The intersection of two events A and B (denoted A  B or “A and B”) is the event consisting of all outcomes in the sample space S that are contained in both event A and event B.
 may be read as “and” since both events occur. This is a joint probability.
(Figure 5.6)

35. Rules of Probability Intersection of Two Events
For example, randomly choose a card from a deck of 52 playing cards.
If Q is the event that we draw a queen and R is the event that we draw a red card, what is Q  R?
It is the possibility of getting both a queen and a red card (2 ways).

36. Rules of Probability General Law of Addition
The general law of addition states that the probability of the union of two events A and B is:
P(A  B) = P(A) + P(B) – P(A  B)
So, you have to subtract P(A  B) to avoid over-stating the probability.
When you add the P(A) and P(B) together, you count the P(A and B) twice.
A and B A B

37. Rules of Probability General Law of Addition For the card example:
P(Q) = 4/52 (4 queens in a deck)
P(R) = 26/52 (26 red cards in a deck)
P(Q  R) = 2/52 (2 red queens in a deck)
P(Q  R) = P(Q) + P(R) – P(Q  Q)
Q and R = 2/52
= 4/ /52 – 2/52
Q 4/52
R 26/52
= 28/52 = or 53.85%

38. Rules of Probability Mutually Exclusive Events Special Law of Addition
Events A and B are mutually exclusive (or disjoint) if their intersection is the null set () that contains no elements.
If A  B = , then P(A  B) = 0
Special Law of Addition
In the case of mutually exclusive events, the addition law reduces to:
P(A  B) = P(A) + P(B)

39. Rules of Probability Collectively Exhaustive Events
Events are collectively exhaustive if their union is the entire sample space S.
Two mutually exclusive, collectively exhaustive events are dichotomous (or binary) events.
For example, a car repair is either covered by the warranty (A) or not (B).
No Warranty
Warranty

40. Rules of Probability Collectively Exhaustive Events
More than two mutually exclusive, collectively exhaustive events are polytomous events.
Cash
Debit
Card
Credit
Card
Check
For example, a Wal-Mart customer can pay by credit card (A), debit card (B), cash (C) or check (D).

41. Rules of Probability Categorical Data
Categorical data can be made dichotomous (binary) by defining the second category as everything not in the first category.
Categorical Data
Binary (Dichotomous) Variable
Vehicle type (SUV, sedan, truck, motorcycle)
X = 1 if SUV, 0 otherwise
A randomly-chosen NBA player’s height
X = 1 if height exceeds 7 feet, 0 otherwise
Tax return type (single, married filing jointly, married filing separately, head of household, qualifying widower)
X = 1 if single, 0 otherwise

42. Rules of Probability Conditional Probability
The probability of event A given that event B has occurred.
Denoted P(A | B). The vertical line “ | ” is read as “given.”
for P(B) > 0 and undefined otherwise

43. Rules of Probability Conditional Probability
Consider the logic of this formula by looking at the Venn diagram.
The sample space is restricted to B, an event that has occurred.
A  B is the part of B that is also in A.
The ratio of the relative size of A  B to B is P(A | B).
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44. Rules of Probability Example: High School Dropouts
Of the population aged 16 – 21 and not in college:
Unemployed
13.5%
High school dropouts
29.05%
Unemployed high school dropouts
5.32%
What is the conditional probability that a member of this population is unemployed, given that the person is a high school dropout?

45. Rules of Probability Example: High School Dropouts First define
U = the event that the person is unemployed
D = the event that the person is a high school dropout
P(U) = .1350
P(D) = .2905
P(UD) = .0532
or 18.31%
P(U | D) = > P(U) = .1350
Therefore, being a high school dropout is related to being unemployed.

46. Independent Events
Event A is independent of event B if the conditional probability P(A | B) is the same as the marginal probability P(A).
To check for independence, apply this test:
If P(A | B) = P(A) then event A is independent of B.
Another way to check for independence:
If P(A  B) = P(A)P(B) then event A is independent of event B since
P(A | B) = P(A  B) = P(A)P(B) = P(A)
P(B) P(B)

47. Independent Events Example: Television Ads
Out of a target audience of 2,000,000, ad A reaches 500,000 viewers, B reaches 300,000 viewers and both ads reach 100,000 viewers.
What is P(A | B)?
.3333 or 33%

48. Independent Events Example: Television Ads
So, P(ad A) = P(ad B) = P(A  B) = P(A | B) = .3333
Are events A and B independent?
P(A | B) = ≠ P(A) = .25
P(A)P(B)=(.25)(.15)=.0375 ≠ P(A  B)=.05

49. Independent Events Dependent Events
When P(A) ≠ P(A | B), then events A and B are dependent.
For dependent events, knowing that event B has occurred will affect the probability that event A will occur.
Statistical dependence does not prove causality.
For example, knowing a person’s age would affect the probability that the individual uses text messaging but causation would have to be proven in other ways.

50. Independent Events Using Actuarial Data
An actuary studies conditional probabilities empirically, using - accident statistics - mortality tables - insurance claims records
Many businesses rely on actuarial services, so a business student needs to understand the concepts of conditional probability and statistical independence.

51. Independent Events Multiplication Law for Independent Events
The probability of n independent events occurring simultaneously is:
P(A1  A2  ...  An) = P(A1) P(A2) ... P(An)
if the events are independent
To illustrate system reliability, suppose a Web site has 2 independent file servers. Each server has 99% reliability. What is the total system reliability? Let,
F1 be the event that server 1 fails
F2 be the event that server 2 fails

52. Independent Events Multiplication Law for Independent Events
Applying the rule of independence:
P(F1  F2 ) =
P(F1) P(F2) =
(.01)(.01) = .0001
So, the probability that both servers are down is
The probability that at least one server is “up” is:
= or 99.99%

53. Independent Events Example: Space Shuttle
Redundancy can increase system reliability even when individual component reliability is low.
NASA space shuttle has three independent flight computers (triple redundancy).
Each has an unacceptable .03 chance of failure (3 failures in 100 missions).
Let Fj = event that computer j fails.

54. Independent Events Example: Space Shuttle
What is the probability that all three flight computers will fail?
P(all 3 fail) = P(F1  F2  F3)
= P(F1) P(F2) P(F3)
 presuming that failures are independent
= (0.03)(0.03)(0.03)
= or 27 in 1,000,000 missions.

55. Independent Events The Five Nines Rule How high must reliability be?
Public carrier-class telecommunications data links are expected to be available % of the time.
The five nines rule implies only 5 minutes of downtime per year.
This type of reliability is needed in many business situations.

56. Independent Events
The Five Nines Rule
For example,

57. Independent Events How Much Redundancy is Needed?
Suppose a certain network Web server is up only 94% of the time. What is the probability of it being down?
P(down) = 1 – P(up) = 1 – .94 = .06
How many independent servers are needed to ensure that the system is up at least 99.99% of the time (or down only = or .01% of the time)?

58. Independent Events How Much Redundancy is Needed?
2 servers: P(F1  F2) = (0.06)(0.06) =
3 servers: P(F1  F2  F3) = (0.06)(0.06)(0.06) =
4 servers: P(F1  F2  F3  F4) = (0.06)(0.06)(0.06)(0.06) =
So, to achieve a 99.99% up time, 4 redundant servers will be needed.

59. Applied Statistics in Business & Economics
End of Chapter 5A
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