Overview Created by Tom Wegleitner Centreville Virginia Edited

Overview 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 2

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

Random Variables Created by Tom Wegleitner, Centreville, Virginia Edited by Olga Pilipets, San Diego,

Definitions v 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 5

Definitions v 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 v 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 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 6

Example 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 7

Definitions v 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 8

Example 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. x 0 1 2 3 P(x) 1/8 3/8 1/8 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 9

Example: # 6 p. 204 Identify Discrete and continuous Random Variables a) The cost of conducting a genetics experiment. 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. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 10

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

Example: #8 p. 204 Identify Probability Distributions. 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. x 0 1 2 3 4 P(x) 0. 502 0. 365 0. 098 0. 011 0. 001 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 12

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

Example: #8 p. 204 Find the mean and standard deviation of the probability distribution. x 0 1 2 3 4 P(x) 0. 502 0. 365 0. 109 0. 023 0. 001 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 14

Roundoff Rule for 2 µ, , and 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 15

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) 0 1 2 3 P(x) 0+ 0+ x (Mexican. Americans) P(x) 7 8 9 4 5 6 0. 001 0. 003 0. 016 10 11 12 0. 053 0. 133 0. 236 0. 283 0. 206 Copyright © 2007 Pearson 0. 069 Education, Inc Publishing as Pearson Addison-Wesley. Slide 16

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

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 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 18

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. v Unusually high: x successes among n trials is an unusually high number of successes if P(x or more) ≤ 0. 05. v Unusually low: x successes among n trials is an unusually low number of successes if P(x or Copyright © 2007 Pearson fewer) ≤ 0. 05. Education, Inc Publishing as Pearson Addison-Wesley. Slide 19

Example: #12 p 204. Identify an unusual event � 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: 0, 1, 2, 3, or 4 and the corresponding probabilities are 0. 0016, 0. 0250, 0. 1432, 0. 3892, and 0. 4096 Is it unusual to randomly select four consumers and find that none of them recognize the brand name of Sony? Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 20

Definition 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)]. E = [x • P(x)] Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 21

Example: #18 p 206. Expected Value in Casino Dice 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 22

Example: #18 p 206. Expected Value for a life insurance Policy The CAN Insurance Company charges a 21 -yearold male a premium of $250 for a one-year $100, 000 life insurance policy. A 21 -yearold male has a 0. 9985 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 23

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