Probability

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 • 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 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? • 1/36 + 2/36+ 1+36 = 4/36 or 11.1% • 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 • (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?

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. -3σ -2σ -1σ +1σ +2σ +3σ μ The Normal Distribution 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% | 95.44% | | 99.72% |

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

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, designateds • 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.)