Chapter 5 Discrete Probability Distributions 1 Copyright © 2012 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

C H A P T E R Outline 5 Discrete Probability Distributions 5-1Probability Distributions 5-2Mean, Variance, Standard Deviation, and Expectation 5-3The Binomial Distribution 5-4Other Types of Distributions (Optional)

C H A P T E R Objectives 5 Discrete Probability Distributions 1Construct a probability distribution for a random variable. 2Find the mean, variance, standard deviation, and expected value for a discrete random variable. 3Find the exact probability for X successes in n trials of a binomial experiment. 4Find the mean, variance, and standard deviation for the variable of a binomial distribution.

C H A P T E R Objectives 5 Discrete Probability Distributions 5Find probabilities for outcomes of variables, using the Poisson, hypergeometric, and multinomial distributions.

5.1 Probability Distributions In Ch. 4, we learned how to find the probabilities that certain events will happen. This information can be used to help us make wiser decisions. In Ch. 5, we will start working our way back to the wonderful, wonderful world of statistics. Bluman, Chapter 55

5.1 Probability Distributions random variable A random variable is a variable whose values are determined by chance. Discrete variable- values can be counted (# of times something happens, etc.) Continuous Variable- can assume an infinitely uncountable amount of values (heights, weight, time, etc.) Ch. 5 is about discrete random variables. 6 Bluman Chapter 5

5.1 Probability Distributions discrete probability distribution A discrete probability distribution consists of the…  Values a random variable can assume  Corresponding probabilities of the values This can be used to create a graphical aid to visualize the different probabilities of each event occurring. Bluman, Chapter 57

Characteristics of Discrete Probability Distribution One row/column of possible data values: X’s One row/column of corr. probabilities: P(X) probabilities The sum of the probabilities of all events in a sample space add up to 1. Each probability is between 0 and 1, inclusively. Prob. can be determined theoretically or based on observation. Bluman, Chapter 58

Chapter 5 Discrete Probability Distributions Section 5-1 Example 5-1 Page #254 9 Bluman Chapter 5

Example 5-1: Rolling a Die Construct a probability distribution for rolling a single die. Bluman, Chapter 510

Example 5-1: Rolling a Die Construct a probability distribution for rolling a single die. 11 Bluman Chapter 5

Chapter 5 Discrete Probability Distributions Section 5-1 Example 5-2 Page # Bluman Chapter 5

Example 5-2: Tossing Coins Represent graphically the probability distribution for the sample space for tossing three coins. Bluman, Chapter 513

Example 5-2: Tossing Coins Represent graphically the probability distribution for the sample space for tossing three coins.. 14 Bluman Chapter 5

Practice Makes Perfect… Example 5-4. #27, page 259. Bluman, Chapter 515

5.2 Introduction 5.1 explains what in the world a probability distribution is. 5.2 begins to inform us how to apply this knowledge practically (a.k.a. statistically). MEANVARIANCE STANDARD DEVIATION EXPECTED VALUE  We will be computing the MEAN, VARIANCE, STANDARD DEVIATION, and the EXPECTED VALUE of discrete probability distributions. Bluman, Chapter 516

5-2 Mean, Variance, Standard Deviation, and Expectation The calculations for these measures are different for probability distributions that we used in Chapter 3. Theoretical VS data set. Consider coin example… Bluman, Chapter 517

5-2 Mean, Variance, Standard Deviation, and Expectation 18 Bluman Chapter 5 In words: Multiply each value by its probability and add all of these up.

Rounding Rule The mean, variance, and standard deviation should be rounded to one more decimal place than the outcome X. When fractions are used, they should be reduced to lowest terms. Mean, Variance, Standard Deviation, and Expectation 19 Bluman Chapter 5

Chapter 5 Discrete Probability Distributions Section 5-2 Example 5-5 Page # Bluman Chapter 5

Example 5-5: Rolling a Die Find the mean of the number of spots that appear when a die is tossed. Bluman, Chapter 521

Example 5-5: Rolling a Die Find the mean of the number of spots that appear when a die is tossed.. 22 Bluman Chapter 5 What does this mean? (Pun intended)

Example 5-7, page 261 If three coins are tossed, find the mean of the number of tails that occur. Bluman, Chapter 523

Chapter 5 Discrete Probability Distributions Section 5-2 Example 5-8 Page # Bluman Chapter 5

Example 5-8: Trips of 5 Nights or More The probability distribution shown represents the number of trips of five nights or more that American adults take per year. (That is, 6% do not take any trips lasting five nights or more, 70% take one trip lasting five nights or more per year, etc.) Find the mean.. 25 Bluman Chapter 5

Example 5-8: Trips of 5 Nights or More 26 Bluman Chapter 5

Variance and Standard Deviation Bluman, Chapter 527 In English: (1)Square each data value. (2)Mult. each data value squared by its probability. (3)Add these up. (4)Subtract the mean squared from this. STANDARD DEVIATION:

Chapter 5 Discrete Probability Distributions Section 5-2 Example 5-9 Page # Bluman Chapter 5

Example 5-9: Rolling a Die Compute the variance and standard deviation for the probability distribution in Example 5–5. Bluman, Chapter 529

Example 5-9: Rolling a Die Compute the variance and standard deviation for the probability distribution in Example 5– Bluman Chapter 5

Chapter 5 Discrete Probability Distributions Section 5-2 Example 5-11 Page # Bluman Chapter 5

Example 5-11: On Hold for Talk Radio A talk radio station has four telephone lines. If the host is unable to talk (i.e., during a commercial) or is talking to a person, the other callers are placed on hold. When all lines are in use, others who are trying to call in get a busy signal. The probability that 0, 1, 2, 3, or 4 people will get through is shown in the distribution. Find the variance and standard deviation for the distribution. 32 Bluman Chapter 5

Example 5-11: On Hold for Talk Radio 33 Bluman Chapter 5

Example 5-11: On Hold for Talk Radio A talk radio station has four telephone lines. If the host is unable to talk (i.e., during a commercial) or is talking to a person, the other callers are placed on hold. When all lines are in use, others who are trying to call in get a busy signal. Should the station have considered getting more phone lines installed? 34 Bluman Chapter 5

Example 5-11: On Hold for Talk Radio No, the four phone lines should be sufficient. The mean number of people calling at any one time is 1.6. Since the standard deviation is 1.1, most callers would be accommodated by having four phone lines because µ + 2  would be (1.1) = = 3.8. Very few callers would get a busy signal since at least 75% of the callers would either get through or be put on hold. (See Chebyshev’s theorem in Section 3–2.) 35 Bluman Chapter 5

Expectation expected value expectation The expected value, or expectation, of a discrete random variable of a probability distribution is the theoretical average of the variable. The expected value is, by definition, the mean of the probability distribution. 36 Bluman Chapter 5

Chapter 5 Discrete Probability Distributions Section 5-2 Example 5-13 Page # Bluman Chapter 5

Example 5-13: Special Die A special six-sided die is made in which 3 sides have 6 spots, 2 sides have 4 spots, and 1 side has 1 spot. If the die is rolled, find the expected value of the number of spots that will occur. 38 Bluman Chapter 5

Practice, Practice, Practice… Example 5-12, page 264. Bluman, Chapter 539

5-3 The Binomial Distribution Many types of probability problems have only two possible outcomes or they can be reduced to two outcomes. Examples include: when a coin is tossed it can land on heads or tails, when a baby is born it is either a boy or girl, when a free throw is shot it can either be made or missed, etc. 40 Bluman Chapter 5

The Binomial Distribution binomial experiment The binomial experiment is a probability experiment that satisfies these requirements: 1.Each trial can have only two possible outcomes—success or failure. 2.There must be a fixed number of trials. 3.The outcomes of each trial must be independent of each other. 4.The probability of success must remain the same for each trial. 41 Bluman Chapter 5

Notation for Binomial Distribution (Pg. 271) The symbol for the probability of success The symbol for the probability of failure The numerical probability of success The numerical probability of failure and P(F) = 1 – p = q The number of trials The number of successes P(S)P(S) P(F)P(F) p q P(S) = p n X Note that X = 0, 1, 2, 3,..., n 42 Bluman Chapter 5

The Binomial Distribution In a binomial experiment, the probability of exactly X successes in n trials is 43 Bluman Chapter 5 These are really awesome, but we will never use them.

Other Ways to Calculate Table B, page (this is really awesome, but not practical). TI-83/84  Steps on page 281  Binompdf(n,p,X) OR binomcdf(n,p,X)  This way is really awesome and practical and easy! It’s just flat out amazing. Bluman, Chapter 544

Example 5-15, page 272 A coin is tossed three times. Find the probability of getting exactly two heads. Now toss the coin 50 times. What is the probability of getting at least 33 tails???? Bluman, Chapter 545

Chapter 5 Discrete Probability Distributions Section 5-3 Example 5-18 Page # Bluman Chapter 5

Example 5-18: Tossing Coins A coin is tossed 3 times. Find the probability of getting exactly two heads, using Table B. 47 Bluman Chapter 5

Chapter 5 Discrete Probability Distributions Section 5-3 Example 5-16 Page # Bluman Chapter 5

Example 5-16: Survey on Doctor Visits A survey found that one out of five Americans say he or she has visited a doctor in any given month. If 10 people are selected at random, find the probability that exactly 3 will have visited a doctor last month. Then, at least 3. Bluman, Chapter 549

Example 5-16: Survey on Doctor Visits A survey found that one out of five Americans say he or she has visited a doctor in any given month. If 10 people are selected at random, find the probability that exactly 3 will have visited a doctor last month. 50 Bluman Chapter 5

Chapter 5 Discrete Probability Distributions Section 5-3 Example 5-17 Page # Bluman Chapter 5

Example 5-17: Survey on Employment A survey from Teenage Research Unlimited (Northbrook, Illinois) found that 30% of teenage consumers receive their spending money from part- time jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them will have part- time jobs. Bluman, Chapter 552

Example 5-17: Survey on Employment A survey from Teenage Research Unlimited (Northbrook, Illinois) found that 30% of teenage consumers receive their spending money from part-time jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them will have part-time jobs. 53 Bluman Chapter 5

The Binomial Distribution The mean, variance, and standard deviation of a variable that has the binomial distribution can be found by using the following formulas. 54 Bluman Chapter 5

Example 5-22, page 275 A die is rolled 480 times. Find the mean, variance, and standard deviation of the number of 3’s that will be rolled. Bluman, Chapter 555

Chapter 5 Discrete Probability Distributions Section 5-3 Example 5-23 Page # Bluman Chapter 5

Example 5-23: Likelihood of Twins The Statistical Bulletin published by Metropolitan Life Insurance Co. reported that 2% of all American births result in twins. If a random sample of 8000 births is taken, find the mean, variance, and standard deviation of the number of births that would result in twins. Bluman, Chapter 557

Example 5-23: Likelihood of Twins The Statistical Bulletin published by Metropolitan Life Insurance Co. reported that 2% of all American births result in twins. If a random sample of 8000 births is taken, find the mean, variance, and standard deviation of the number of births that would result in twins. 58 Bluman Chapter 5