Download Probability

Learning Objectives
In this chapter you will learn
the basic rules of probability
about estimating the probability of
the occurrence of an event
the Central Limit Theorem
how to establish confidence
intervals
Types of Probability
Three approaches to probability
Mathematical
Empirical
Subjective
Mathematical Probability
Mathematical (or classical) probability
based on equally likely outcomes that
can be calculated
useful when equal chance of
outcomes and random selection is
possible
Example
20 people are arrested for crimes
2 are innocent
If one of the accused is picked
randomly, what is the probability of
selecting and innocent person?
Solution
2/20 or .1 – 10% chance of picking an innocent
person
Empirical Probability
Empirical probability
uses the frequency of past events
to predict the future
calculated
the number of times an event occurred in
the past
divided by the number of observations
Example
75,000 autos were registered in
the county last year
650 were reported stolen
What is the probability of having a
car stolen this year?
Solution
650/75,000  .009 or .9%
Subjective Probability
Subjective probability
based on personal reflections of an
individual’s opinion about an event
used when no other information is
available
Example
What is the probability that Al
Gore will win the next presidential
election?
 Obviously, the answer depends on
who you ask!
Probability Rules
We sometimes need to combine
the probability of events
 two fundamental methods of
combining probabilities are
by addition
by multiplication
The Addition Rule
The Addition Rule
if two events are mutually exclusive
(cannot happen at the same time)
the probability of their occurrence is
equal to the sum of their separate
probabilities
P(A or B) = P(A) + P(B)
Example
What is the probability that an
odd number will result from the
roll of a single die?
6 possible outcomes, 3 of which
are odd numbers
1 1 1 1
Formula
    .50
6
6
6
2
The Multiplication Rule
Suppose that we want to find
the probability of two (or more events)
occurring together?
The Multiplication Rule
probability of events are NOT mutually
exclusive
equals the product of their separate
probabilities
P(B|A) = P(A) times P(B|A)
Example
Two cards are selected, without
replacement, from a standard deck
What is probability of selecting a 10
and a 4?
P(B|A) = P(A) times P(B|A)
4
4
16


 .006
52 51 2652
Laws of Probability
The probability that an event will occur
is equal to the ratio of “successes” to the
number of possible outcomes
the probability that you would flip a coin
that comes up “heads” is one out of two or
50%
Gambler’s Fallacy
Probability of flipping a head
extends to the next toss and every toss
thereafter
mistaken belief that
if you tossed ten heads in a row
the probability of tossing another is
astronomical
in fact, it has never changed – it is still
50%
Calculating Probability
You can calculate the probability
of any given total that can be
thrown in a game of “Craps”
each die has 6 sides
when a pair of dice is thrown,
there are how many possibilities?
Outcomes of Rolling Dice
Die #1 Roll
Die #2
Roll
1
2
3
4
5
6
1
2
3
4
5
6
2
3
4
5
6
7
3
4
5
6
7
8
4
5
6
7
8
9
5
6
7
8
9
10
6
7
8
9
10
11
7
8
9
10
11
12
Number of Ways to Roll Each Total
Total Roll
2
3
4
5
6
7
8
9
10
11
12
N of Ways
1
2
3
4
5
6
5
4
3
2
1
Winning or Not?
What is the probability of….
losing on the first roll?
1/36 + 2/36+ 1+36 = 4/36 or
11.1%
rolling a ten?
3/36 or 1/12 = 8.3%
Next Roll
making the point on the next roll?
now we calculate probability
P(10) + P(any number, any roll) = 1/3
(1/12) times (1/3) = 2.8%
Making the Point
The probability of making the point for
any number
to calculate this probability
use both the Addition Rule and the
Multiplication Rule
the probability of two events that are not mutually
exclusive are the product of their separate
probability
Continuing
Add the separate probabilities of rolling
each type of number
P(10) x P (any number, any roll) = 1/12 x 1/3
= 1/36 or 2.8% is the P of two 10s or two 4s
P of two 5s or two 9s = (1/9) (2/5) = 2/45 =
4.4%
P of two 6s or two 8s = (5/36) (5/11) =
25/396 = 6.3%
Who Really Wins?
Add up all the probabilities of winning
(2/9) + 2 (1/36) + 2 (2/45) + 2
(25/396) = (2/9) + (4/45) + (25/198)
= 244/495 or 49.3%
What is the probability that you will
lose in the long run or that the
Casino wins?
Empirical Probability
Empirical probability is based upon
research findings
Example: Study of Victimization
Rates among American Indians
Which group had the greatest rate
of violent crime victimizations?
The lowest rate?
Violent Crime Victimization By Age,
Race, & Sex of Victim, 1992 - 1996
Percent of Violent Crime Victimization
Highest rate
byAmerican
race & age
Victim
Age/Sex
Indian
12 – 17
20.4%
18 – 24
31.5
25 – 34
23.5
35 – 44
18.0
45 – 54
4.7
55 & Older
1.9
MALE
58.9
FEMALE Lowest
41.1rate
by race & age
White
23.8%
23.4
23.6
17.1
7.8
4.3
58.4
41.6
Black
26.8%
24.0
23.2
16.6
6.1
3.3
50.5
49.5
Asian
24.0%
21.7
26.3
18.3
7.3
2.4
62.6
37.4
Total
24.2%
23.6
23.6
17.0
7.5
4.1
57.4
42.6
Using Probability
We use probability every day
statements such as
will it may rain today?
will the Red Sox win the World Series?
will someone break into my house?
We use a model to illustrate probability
the normal distribution
The Normal Distribution
Approximately 68% of area
under the curve falls with 
1 standard deviation from
68.26% the mean
Approximately
1.5% of area
falls beyond 
3 standard
deviations
|
|
95.44%
99.72%
|
|
-3σ -2σ -1σ μ +1σ +2σ +3σ
Z Scores
The standard score, or z-score
represents the number of standard
deviations
a random variable x falls from the
mean μ
value - mean
x
z

standard deviation

Example
The mean of test scores is 95 and
the standard deviation is 15
find the z-score for a person who
scored an 88
Solution
88  95
 0.467
15
Example Continued
We then convert the z-score into
the area under the curve
look at Appendix A.2 in the text
the fist column is the first & second
values of z (0.4)
the top row is the third value (6)
cumulative area = .3228
Another Use of Probability
We can also take advantage of
probability when we draw samples
social scientists like the properties
of the normal distribution
the Central Limit Theorem is
another example of probability
The Central Limit Theorem
If repeated random samples
of a given size are drawn
from any population (with a mean of 
and a variance of )
then as the sample size becomes large
the sampling distribution of sample
means approaches normality
Example
Dot/Lines
15
10
Count
Roll a pair of dice
100 times
The shape of the
distribution of
outcomes will
resemble this
figure
5
0
2.5
5.0
7.5
v1
10.0
Standard Error of the
Sample Means
The standard error of the sample
means
is the standard deviation
of the sampling distribution of the
sample means

x 
n
Standard Error of the
Sample Means
If  is not known and n  30
the standard deviation of the
sample, designated s
is used to approximate the
population standard deviation
the formula for the standard error
then becomes:
s
sx 
n
Confidence Intervals
An Interval Estimate states the range
within which a population parameter
probably lies
the interval within which a population
parameter is expected to occur is called
a confidence interval
two confidence intervals commonly
used are the 95% and the 99%
Constructing Confidence
Intervals
In general, a confidence interval
for the mean is computed by:
s
X Z
n
95% and 99%
Confidence Intervals
95% CI for the population mean is
calculated by
s
X  1.96
n
99% CI for the population mean is
calculated by
s
X  2.58
n
Summary
Social scientists use probability
to calculate the likelihood that an
event will occur
in various combinations
for various purposes (estimating a
population parameter, distribution
of scores, etc.)