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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.)