# Probability.

## Presentation on theme: "Probability."— Presentation transcript:

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
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
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

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 Formula

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)

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

Winning or Not? What is the probability of…. losing on the first roll?
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
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
Highest rate by race & age Lowest rate by race & age

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
μ 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

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

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

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

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

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:

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:

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

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.)