Slide Slide 1 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Created by Tom Wegleitner, Centreville, Virginia Edited by Olga Pilipets, San Diego, California Overview

Slide Slide 2 This chapter will deal with the construction of discrete probability distributions Probability Distributions will describe what will probably happen instead of what actually did happen. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 3 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Combining Descriptive Methods and Probabilities In this chapter we will construct probability distributions by presenting possible outcomes along with the relative frequencies we expect.

Slide Slide 4 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Created by Tom Wegleitner, Centreville, Virginia Edited by Olga Pilipets, San Diego, California Random Variables

Slide Slide 5  Random variable a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 6  Discrete random variable Random variable x takes on counting (natural) values, where “counting” refers to the fact that there might be infinitely many values, but they result from a counting process Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.  Continuous random variable infinitely many values, and those values can be associated with measurements on a continuous scale in such a way that there are no gaps or interruptions

Slide Slide 7 We could flip a coin 3 times. There are 8 possible outcomes; S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} If we are interested in the number of heads that appear during the three flips we could get any of the following numbers: 0, 1, 2, or 3. The numbers 0, 1, 2, and 3, are the values of a random variable that has been associated with the possible outcomes. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 8  Probability distribution a listing of all possible values that the variable can assume along with their corresponding probabilities. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 9 A coin is flipped 3 times giving the following sample space S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} We are interested in the number of heads that appear during the three flips. Assigning a probability value to each possible random variable we construct the following probability distribution. x0123 P(x) Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. 1/8 3/8 1/8

Slide Slide 10 a) The cost of conducting a genetics experiment. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. b) The number of supermodels who ate pizza yesterday. c) The exact life span of a kitten. d) The number of statistic professors who read a newspaper this morning e) The weight of a feather.

Slide Slide 11 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. P ( x ) = 1 where x assumes all possible values.  0  P ( x )  1 for every individual value of x.

Slide Slide 12 A researcher reports that when groups of four children are randomly selected from a population of couples meeting certain criteria, the probability distribution for the number of boys is given in the accompanying table. x01234 P(x) Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 13 µ =  [x P(x)] Mean  2 =  [ (x – µ) 2 P(x )] Variance  2 = [  x 2 P ( x )] – µ 2 Variance (shortcut )  =  [ x 2 P ( x )] – µ 2 Standard Deviation Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Mean, Variance and Standard Deviation of a Probability Distribution

Slide Slide 14 x01234 P(x) Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 15 Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round µ,  and  2 to one decimal place. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 16 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example Twelve jurors are to be randomly selected from a population in which 80% of the jurors are Mexican- American. If we assume that jurors are randomly selected without bias, and if we let x = the number of Mexican-American jurors among 12 jurors, we will get a probability distribution represented by the following table: Cont-d x (Mexican- Americans) P(x) x (Mexican- Americans) P(x)

Slide Slide 17 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Graphs The probability histogram is very similar to a relative frequency histogram, but the vertical scale shows probabilities.

Slide Slide 18 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Identifying Unusual Results Range Rule of Thumb According to the range rule of thumb, most values should lie within 2 standard deviations of the mean. We can therefore identify “unusual” values by determining if they lie outside these limits: Maximum usual value = μ + 2σ Minimum usual value = μ – 2σ

Slide Slide 19 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Identifying Unusual Results Probabilities Rare Event Rule If, under a given assumption (such as the assumption that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is probably not correct.  Unusually high: x successes among n trials is an unusually high number of successes if P(x or more) ≤  Unusually low: x successes among n trials is an unusually low number of successes if P(x or fewer) ≤ 0.05.

Slide Slide 20  In a study of brand recognition of Sony, groups of four consumers are interviewed. If x is the number of people in the group who recognize the Sony brand name, then x can be: Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. 0, 1, 2, 3, or 4 and the corresponding probabilities are , , , , and Is it unusual to randomly select four consumers and find that none of them recognize the brand name of Sony?

Slide Slide 21 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. E =  [x P(x)] The expected value of a discrete random variable is denoted by E, and it represents the average value of the outcomes. It is obtained by finding the value of  [x P(x)].

Slide Slide 22 When you give a casino $5 for a bet on the “pass line” in a casino game of dice, there is a 251/495 probability that you will lose $5 and there is a 244/495 probability that you will make a net gain of $5. (If you win, the casino gives you $5 and you get to keep your $5 bet, so the net gain is $5.) What is your expected value? In the long run, how much do you lose for each dollar bet? Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 23 The CAN Insurance Company charges a 21-year- old male a premium of $250 for a one-year $100,000 life insurance policy. A 21-year- old male has a probability of living for a year. a) From the perspective of a 21-year-old male (or his estate), what are the values of the two different outcomes? b) What is the expected value for a 21-year-old male who buys the insurance? c) What would be the cost of the insurance policy if the company just breaks even (in the long run with many such policies), instead of making a profit? Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

Slide Slide 24 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Recap In this section we have discussed:  Probability histograms.  Requirements for a probability distribution.  Mean, variance and standard deviation of a probability distribution.  Random variables and probability distributions.  Expected value.