Download Probability

5A-1
Chapter
5A
Probability (Part 1)
Random Experiments
Probability
Rules of Probability
Independent Events
McGraw-Hill/Irwin
© 2008 The McGraw-Hill Companies, Inc. All rights reserved.
5A-3
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.
5A-4
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}
5A-5
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:
S = {(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)}
5A-6
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}
5A-7
Random Experiments
 Events
• Consider the random experiment of tossing a
balanced coin.
What is the sample space?
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?
5A-8
Random Experiments
 Events
• 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 = {E1, E2}
B = {E3, E5, E6}
• These are
displayed in a
Venn diagram:
5A-9
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)}
5A-10
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.
5A-11
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%
P(cash) = .35
check:
18%
P(check) = .18
Sum = 100%
Sum = 1.0
P(credit card) = .32
Probability
P(debit card) = .20
5A-12
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.
5A-13
Probability
5A-14
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.
5A-15
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%
5A-16
Rules of Probability
 Odds of an Event
• The odds in favor of event A occurring is
Odds =
P( A)
P( A)

P( A ') 1  P( A)
• 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?
P(rolling seven)
1/ 6
1/ 6 1
Odds =



1  P(rolling seven) 1  1/ 6 5/ 6 5
5A-17
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.
5A-18
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:
a
P(A) =
ab
• For example, if a race horse has a 4 to 1 odds
against winning, the P(win) is
a
1
1

  0.20 or 20%
P(win) =
a  b 4 1 5
5A-19
Rules of Probability
 Union of Two Events
• 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.
5A-20
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).
5A-21
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.
5A-22
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).
5A-23
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)
When you add
So, you have
A and B
the P(A) and
to subtract
P(B) together,
P(A  B) to
you count the
avoid overA
B
P(A and B)
stating the
twice.
probability.
5A-24
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 + 26/52 – 2/52
= 28/52 = .5385 or 53.85%
Q
4/52
R
26/52
5A-25
Rules of Probability
 Mutually Exclusive Events
• 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)
5A-26
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.”
P( A  B)
P( A | B) 
P( B)
for P(B) > 0 and
undefined otherwise
5A-27
Rules of Probability
 Conditional Probability
• Consider the logic of this formula by looking at the
Venn diagram.
The sample space is
P( A  B)
restricted to B, an event
P( A | B) 
P( B)
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).
5A-28
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?
5A-29
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(D) = .2905
P(UD) = .0532
P(U) = .1350
P(U  D) .0532
P(U | D) 

 .1831 or 18.31%
P ( D)
.2905
• P(U | D) = .1831 > P(U) = .1350
• Therefore, being a high school dropout is related
to being unemployed.
5A-30
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)
5A-31
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.
300, 000
500, 000
P( B) 
 .15
P( A) 
 .25
2, 000, 000
2, 000, 000
100, 000
P( A  B) 
 .05
2, 000, 000
• What is P(A | B)?
P( A  B) .05
.3333 or 33%
P( A | B) 

 .30
P( B)
.15
5A-32
Independent Events
 Example: Television Ads
• So, P(ad A) = .25
P(ad B) = .15
P(A  B) = .05
P(A | B) = .3333
• Are events A and B independent?
• P(A | B) = .3333 ≠ P(A) = .25
• P(A)P(B)=(.25)(.15)=.0375 ≠ P(A  B)=.05
5A-33
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.