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Learning ObjectivesIn this chapter you will learn
the basic rules of probabilityabout estimating the probability of the occurrence of an event
the Central Limit Theoremhow to establish confidence intervals
Mathematical Probability
Mathematical (or classical) probabilitybased on equally likely outcomes that can be calculated
useful when equal chance of outcomes and random selection is possible
Example20 people are arrested for crimes 2 are innocentIf one of the accused is picked
randomly, what is the probability of selecting and innocent person?
Solution2/20 or .1 – 10% chance of picking an
innocent person
Empirical ProbabilityEmpirical 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
Example75,000 autos were registered in the county last year650 were reported stolenWhat is the probability of having a car stolen this year?
Solution650/75,000 .009 or .9%
Subjective ProbabilitySubjective probabilitybased on personal reflections of an individual’s opinion about an event
used when no other information is available
ExampleWhat is the probability that Al Gore will win the next presidential election? Obviously, the answer depends on who you ask!
Probability RulesWe sometimes need to combine the probability of events two fundamental methods of combining probabilities areby additionby multiplication
The Addition RuleThe 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)
ExampleWhat 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
Formula 50.2
1
6
1
6
1
6
1==++
The Multiplication Rule
Suppose that we want to find the probability of two (or more events) occurring
together?
The Multiplication Ruleprobability of events are NOT mutually exclusive equals the product of their separate probabilitiesP(B|A) = P(A) times P(B|A)
ExampleTwo cards are selected, without replacement, from a standard deckWhat is probability of selecting a 10 and a 4?
P(B|A) = P(A) times P(B|A)006.
2652
16
51
4
52
4≈=•
Laws of ProbabilityThe probability that an event will
occuris 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 FallacyProbability of flipping a head extends to the next toss and every toss
thereafter mistaken belief that
if you tossed ten heads in a rowthe probability of tossing another is
astronomicalin 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 sideswhen a pair of dice is thrown, there are how many possibilities?
Die #1 Roll
Die #2Roll
1 2 3 4 5 6
1 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 94 5 6 7 8 9 105 6 7 8 9 10 116 7 8 9 10 11 12
Outcomes of Rolling Dice
Number of Ways to Roll Each Total
Total Roll N of Ways 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4
10 3 11 2 12 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 Rollmaking 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 probabilityuse both the Addition Rule and
the Multiplication Rulethe probability of two events that are not
mutually exclusive are the product of their separate probability
ContinuingAdd the separate probabilities of rolling each type of numberP(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 ProbabilityEmpirical probability is based
upon research findingsExample: 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
VictimAge/Sex
AmericanIndian White Black Asian Total
12 – 17 20.4% 23.8% 26.8% 24.0% 24.2%18 – 24 31.5 23.4 24.0 21.7 23.625 – 34 23.5 23.6 23.2 26.3 23.635 – 44 18.0 17.1 16.6 18.3 17.045 – 54 4.7 7.8 6.1 7.3 7.5
55 & Older 1.9 4.3 3.3 2.4 4.1MALE 58.9 58.4 50.5 62.6 57.4
FEMALE 41.1 41.6 49.5 37.4 42.6
Highest rate by race & age
Lowest rate by race & age
Using ProbabilityWe use probability every day
statements such aswill it may rain today?will the Red Sox win the World Series?will someone break into my house?
We use a model to illustrate probabilitythe normal distribution
The Normal Distribution
μ +2σ-2σ +1σ +3σ-1σ-3σ
Approximately 68% of area under the curve falls with 1 standard deviation from the mean68.26%
| 95.44% |
| 99.72% |
Approximately 1.5% of area
falls beyond 3 standard deviations
Z ScoresThe standard score, or z-scorerepresents the number of standard
deviations a random variable x falls from the
mean μ
σμ−
==x
zdeviation standard
mean - value
ExampleThe mean of test scores is 95
and the standard deviation is 15find the z-score for a person who scored an 88
Solution 467.015
9588−≈
−
Example ContinuedWe then convert the z-score into
the area under the curvelook at Appendix A.2 in the textthe 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 samplessocial 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 μ
and a variance of σ)then as the sample size becomes largethe sampling distribution of sample
means approaches normality
ExampleRoll a pair of
dice 100 timesThe shape of
the distribution of outcomes will
resemble this figure
Dot/Lines show counts
2.5 5.0 7.5 10.0
v1
0
5
10
15
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
σσ
x n=
Standard Error of the Sample Means
If σ 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:
ss
nx =
Confidence IntervalsAn Interval Estimate states the
range within which a population parameter probably liesthe 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:
X Zs
n±
95% and 99% Confidence Intervals 95% CI for the population mean is calculated by
Xs
n±196.
Xs
n±258.
99% CI for the population mean is calculated by