1. Probability

2. 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

3. Types of Probability Three approaches to probability Mathematical
Empirical
Subjective

4. 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

5. 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

6. 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

7. 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%

8. Subjective Probability
based on personal reflections of an individual’s opinion about an event
used when no other information is available

9. Example
What is the probability that Al Gore will win the next presidential election?
Obviously, the answer depends on who you ask!

10. Probability Rules
We sometimes need to combine the probability of events
two fundamental methods of combining probabilities are
by addition
by multiplication

11. 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)

12. 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
Formula

13. 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)

14. 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)

15. 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%

16. 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%

17. 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?

18. Outcomes of Rolling Dice

20. Winning or Not? What is the probability of…. losing on the first roll?
rolling a ten?
3/36 or 1/12 = 8.3%

21. 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%

22. 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

23. 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%

24. Who Really Wins? Add up all the probabilities of winning
What is the probability that you will lose in the long run or that the Casino wins?

25. 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?

26. Violent Crime Victimization By Age, Race, & Sex of Victim, 1992 - 1996
Highest rate
by race & age
Lowest rate by race & age

27. 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

28. The Normal Distributionμ
Approximately 68% of area under the curve falls with  1 standard deviation from the mean
Approximately 1.5% of area falls beyond  3 standard deviations
68.26%
| % |
| % |
+2σ
-2σ
+1σ
+3σ
-1σ
-3σ

29. Z Scores The standard score, or z-score
represents the number of standard deviations
a random variable x falls from the mean μ

30. 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

31. 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

32. 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

33. 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

34. Example Roll a pair of dice 100 times
The shape of the distribution of outcomes will resemble this figure

35. 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

36. 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:

37. 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%

38. Constructing Confidence Intervals
In general, a confidence interval for the mean is computed by:

39. 95% and 99% Confidence Intervals
95% CI for the population mean is calculated by
99% CI for the population mean is calculated by

40. 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.)