1 5-1 Review and Preview 5-2 Random Variables 5-3 Binomial Probability Distributions 5-4 Mean, Variance, and Standard Deviation for the Binomial Distribution 5-5 Poisson Probability Distributions Discrete Probability Distributions 202
2 CHAPTER PROBLEM Did Mendel s results from plant hybridization experiments contradict his theory? Gregor Mendel conducted original experiments to study the genetic traits of pea plants. In 1865 he wrote Experiments in Plant Hybridization, which was published in Proceedings of the Natural History Society. Mendel presented a theory that when there are two inheritable traits, one of them will be dominant and the other will be recessive. Each parent contributes one gene to an offspring and, depending on the combination of genes, that offspring could inherit the dominant trait or the recessive trait. Mendel conducted an experiment using pea plants. The pods of pea plants can be green or yellow. When one pea carrying a dominant green gene and a recessive yellow gene is crossed with another pea carrying the same green> yellow genes, the offspring can inherit any one of four combinations of genes, as shown in the table below. Because green is dominant and yellow is recessive, the offspring pod will be green if either of the two inherited genes is green. The offspring can have a yellow pod only if it inherits the yellow gene from each of the two parents. We can see from the table that when crossing two parents with the green> yellow pair of genes, we expect that 3> 4 of the offspring peas should have green pods. That is, P(green pod) = 3>4. When Mendel conducted his famous hybridization experiments using parent pea plants with the green> yellow combination of genes, he obtained 580 offspring. According to Mendel s theory, 3>4 of the offspring should have green pods, but the actual number of plants with green pods was 428. So the proportion of offspring with green pods to the total number of offspring is 428>580 = Mendel expected a proportion of 3>4 or 0.75, but his actual result is a proportion of In this chapter we will consider the issue of whether the experimental results contradict the theoretical results and, in so doing, we will lay a foundation for hypothesis testing, which is introduced in Chapter 8. Gene from Parent 1 Gene from Parent 2 Offspring Genes Color of Offspring Pod green + green : green> green : green green + yellow : green> yellow : green yellow + green : yellow> green : green yellow + yellow : yellow> yellow : yellow
3 204 Chapter 5 Discrete Probability Distributions 5-1 Review and Preview In this chapter we combine the methods of descriptive statistics presented in Chapters 2 and 3 and those of probability presented in Chapter 4 to describe and analyze probability distributions. Probability distributions describe what will probably happen instead of what actually did happen, and they are often given in the format of a graph, table, or formula. Recall that in Chapter 2 we used observed sample data to construct frequency distributions. In this chapter we use the possible outcomes of a procedure (determined using the methods of Chapter 4) along with the expected relative frequencies to construct probability distributions, which serve as models of theoretically perfect frequency distributions. With this knowledge of population outcomes, we are able to find important characteristics, such as the mean and standard deviation, and to compare theoretical probabilities to actual results in order to determine whether outcomes are unusual. Figure 5-1 provides a visual summary of what we will accomplish in this chapter. Using the methods of Chapters 2 and 3, we would repeatedly roll the die to collect sample data, which could then be described visually (with a histogram or boxplot), or numerically using measures of center (such as the mean) and measures of variation (such as the standard deviation). Using the methods of Chapter 4, we could find the probability of each possible outcome. Then we could construct a probability distribution, which describes the relative frequency table for a die rolled an infinite number of times. In order to fully understand probability distributions, we must first understand the concept of a random variable, and be able to distinguish between discrete and continuous random variables. In this chapter we focus on discrete probability distributions. In particular, we discuss binomial and Poisson probability distributions. We will discuss continuous probability distributions in Chapter 6. The table at the extreme right in Figure 5-1 represents a probability distribution that serves as a model of a theoretically perfect population frequency distribution. Chapters 2 and 3 Collect sample data, then get statistics and graphs. x f x 3.6 s 1.7 Chapter 5 Create a theoretical model describing how the experiment is expected to behave, then get its parameters. Roll a die Chapter 4 Find the probability for each outcome. P (1) 1/6 P (2) 1/6 P (6) 1/6 x P(x) 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/ Figure 5-1 Combining Descriptive Methods and Probabilities to Form a Theoretical Model of Behavior
4 5-2 Random Variables 205 In essence, we can describe the relative frequency table for a die rolled an infinite number of times. With this knowledge of the population of outcomes, we are able to find its important characteristics, such as the mean and standard deviation. The remainder of this book and the very core of inferential statistics are based on some knowledge of probability distributions. We begin by examining the concept of a random variable, and then we consider important distributions that have many real applications. 5-2 Random Variables Key Concept In this section we consider the concept of random variables and how they relate to probability distributions. We also discuss how to distinguish between discrete random variables and continuous random variables. In addition, we develop formulas for finding the mean, variance, and standard deviation for a probability distribution. Most importantly, we focus on determining whether outcomes are likely to occur by chance or they are unusual (in the sense that they are not likely to occur by chance). We begin with the related concepts of random variable and probability distribution. A random variable is a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure. A probability distribution is a description that gives the probability for each value of the random variable. It is often expressed in the format of a graph, table, or formula. Table 5-1 Probability Distribution: Probabilities of Numbers of Peas with Green Pods Among 5 Offspring Peas x (Number of Peas with Green Pods) P(x) Genetics Consider the offspring of peas from parents both having the green> yellow combination of pod genes. Under these conditions, the probability that the offspring has a green pod is 3>4 or That is, P(green) = If five such offspring are obtained, and if we let x = number of peas with green pods among 5 offspring peas then x is a random variable because its value depends on chance. Table 5-1 is a probability distribution because it gives the probability for each value of the random variable x. (In Section 5-3 we will see how to find the probability values, such as those listed in Table 5-1.) Note: If a probability value is very small, such as , we can represent it as 0+ in a table, where 0+ indicates that the probability value is a very small positive number. (Representing the small probability as 0 would incorrectly indicate that the event is impossible.) In Section 1-2 we made a distinction between discrete and continuous data. Random variables may also be discrete or continuous, and the following two definitions are consistent with those given in Section 1-2.
5 206 Chapter 5 Discrete Probability Distributions Figure 5-2 Devices Used to Count and Measure Discrete and Continuous Random Variables Counter Graph of Discrete Values (a) Discrete Random Variable: Count of the number of movie patrons Voltmeter 0 9 Graph of Continuous Values 0 9 (b) Continuous Random Variable: The measured voltage of a smoke detector battery. A discrete random variable has either a finite number of values or a countable number of values, where countable refers to the fact that there might be infinitely many values, but they can be associated with a counting process, so that the number of values is 0 or 1 or 2 or 3, etc. A continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions. This chapter deals exclusively with discrete random variables, but the following chapters will deal with continuous random variables. 2 The following are examples of discrete and continuous random variables. 1. Discrete Let x = the number of eggs that a hen lays in a day. This is a discrete random variable because its only possible values are 0, or 1, or 2, and so on. No hen can lay eggs, which would have been possible if the data had come from a continuous scale. 2. Discrete The count of the number of statistics students present in class on a given day is a whole number and is therefore a discrete random variable. The counting device shown in Figure 5-2(a) is capable of indicating only a finite number of values, so it is used to obtain values for a discrete random variable. 3. Continuous Let x = the amount of milk a cow produces in one day. This is a continuous random variable because it can have any value over a continuous span. During a single day, a cow might yield an amount of milk that can be any value
6 5-2 Random Variables 207 between 0 gallons and 5 gallons. It would be possible to get gallons, because the cow is not restricted to the discrete amounts of 0, 1, 2, 3, 4, or 5 gallons. 4. Continuous The measure of voltage for a particular smoke detector battery can be any value between 0 volts and 9 volts. It is therefore a continuous random variable. The voltmeter shown in Figure 5-2(b) is capable of indicating values on a continuous scale, so it can be used to obtain values for a continuous random variable. Graphs There are various ways to graph a probability distribution, but we will consider only the probability histogram. Figure 5-3 is a probability histogram. Notice that it is similar to a relative frequency histogram (see Chapter 2), but the vertical scale shows probabilities instead of relative frequencies based on actual sample results. In Figure 5-3, we see that the values of 0, 1, 2, 3, 4, 5 along the horizontal axis are located at the centers of the rectangles. This implies that the rectangles are each 1 unit wide, so the areas of the rectangles are 0.001, 0.015, 0.088, 0.264, 0.396, The areas of these rectangles are the same as the probabilities in Table 5-1. We will see in Chapter 6 and future chapters that such a correspondence between area and probability is very useful in statistics. Every probability distribution must satisfy each of the following two requirements. Requirements for a Probability Distribution 1. P (x) = 1 where x assumes all possible values. (The sum of all probabilities must be 1, but values such as or are acceptable because they result from rounding errors.) 2. 0 P(x) 1 for every individual value of x. (That is, each probability value must be between 0 and 1 inclusive.) The first requirement comes from the simple fact that the random variable x represents all possible events in the entire sample space, so we are certain (with probability 1) that one of the events will occur. In Table 5-1 we see that the sum of the probabilities is (due to rounding errors) and that every value P(x) is between 0 and 1. Because Table 5-1 satisfies the above requirements, we confirm that it is a probability distribution. A probability distribution may be described by a table, such as Table 5-1, or a graph, such as Figure 5-3, or a formula. 0.4 Life Data Analysis Life data analysis deals with the longevity and failure rates of manufactured products. In one application, it is known that Dell computers have an infant mortality rate, whereby the failure rate is highest immediately after the computers are produced. Dell therefore tests or burns-in the computers before they are shipped. Dell can optimize profits by using an optimal burn-in time that identifies failures without wasting valuable testing time. Other products, such as cars, have failure rates that increase over time as parts wear out. If General Motors or Dell or any other company were to ignore the use of statistics and life data analysis, it would run the serious risk of going out of business because of factors such as excessive warranty repair costs or the loss of customers who experience unacceptable failure rates. The Weibull distribution is a probability distribution commonly used in life data analysis applications. That distribution is beyond the scope of this book. Figure 5-3 Probability Histogram Probability Number of Peas with Green Pods Among 5
7 208 Chapter 5 Discrete Probability Distributions Table 5-2 Cell Phones per Household x P(x) Cell Phones Based on a survey conducted by Frank N. Magid Associates, Table 5-2 lists the probabilities for the number of cell phones in use per household. Does Table 5-2 describe a probability distribution? To be a probability distribution, P (x) must satisfy the preceding two requirements. But P (x) = P(0) + P(1) + P (2) + P (3) = = 0.91 [showing that P (x) Z 1] Because the first requirement is not satisfied, we conclude that Table 5-2 does not describe a probability distribution. 4 Does P (x) = a probability distribution? x 10 (where x can be 0, 1, 2, 3, or 4) determine For the given formula we find that P(0) = 0>10, P(1) = 1>10, P(2) = 2>10, P(3) = 3>10, and P(4) = 4>10, so that 1. P (x) = = = 1 2. Each of the P(x) values is between 0 and 1. Because both requirements are satisfied, the formula given in this example is a probability distribution. Mean, Variance, and Standard Deviation In Chapter 2 we described the following characteristics of data (which can be remembered with the mnemonic of CVDOT for Computer Viruses Destroy Or Terminate ): (1) center; (2) variation; (3) distribution; (4) outliers; and (5) time (changing characteristics of data over time). These same characteristics can be used to describe probability distributions. A probability histogram or table can provide insight into the distribution of random variables. The mean is the central or average value of the random variable for a procedure repeated an infinite number of times. The variance and standard deviation measure the variation of the random variable. The mean, variance, and standard deviation for a probability distribution can be found by using these formulas: Formula 5-1 Formula 5-2 Formula 5-3 Formula 5-4 m = [x # P (x)] s 2 = [(x - m) 2 # P(x)] s 2 = [x 2 # P (x)] - m 2 s = 2 [x 2 # P(x)] - m 2 Mean for a probability distribution Variance for a probability distribution (easier to understand) Variance for a probability distribution (easier computations) Standard deviation for a probability distribution
8 5-2 Random Variables Finding the Mean, Variance, and Standard Deviation Table 5-1 describes the probability distribution for the number of peas with green pods among 5 offspring peas obtained from parents both having the green> yellow pair of genes. Find the mean, variance, and standard deviation for the probability distribution described in Table 5-1 from Example 1. In Table 5-3, the two columns at the left describe the probability distribution given earlier in Table 5-1, and we create the three columns at the right for the purposes of the calculations required. Using Formulas 5-1 and 5-2 and the table results, we get Mean: m = [x # P (x)] = = 3.8 (rounded) Variance: s 2 = [(x - m) 2 # P(x)] = = 0.9 (rounded) The standard deviation is the square root of the variance, so Standard deviation: s = = = 1.0 Table 5-3 Calculating M, S, and S 2 for a Probability Distribution x P(x) x P (x) (x M) 2 P (x) (rounded) = ( ) = = ( ) = = ( ) = = ( ) = = ( ) = = ( ) = Total c c m = [x # P(x)] s 2 = [(x - m) 2 # P(x)] The mean number of peas with green pods is 3.8 peas, the variance is 0.9 peas squared, and the standard deviation is 1.0 pea. Rationale for Formulas 5-1 through 5-4 Instead of blindly accepting and using formulas, it is much better to have some understanding of why they work. When computing the mean from a frequency distribution, f represents class frequency and N represents population size. In the expression below, we rewrite the formula for the mean of a frequency table so that it applies to a population. In the fraction f >N, the value of f is the frequency with which the value x occurs and N is the population size, so f >N is the probability for the value of x. When we replace f> N with P (x), we make the transition from relative frequency based on a limited number of observations to probability based on infinitely many trials. m = ( f # x) N # # # # # # = a c f # x N d = a cx # f N d = a [x # P(x)] Similar reasoning enables us to take the variance formula from Chapter 3 and apply it to a random variable for a probability distribution; the result is Formula 5-2. # # # # # # How to Choose Lottery Numbers Many books and suppliers of computer programs claim to be helpful in predicting winning lottery numbers. Some use the theory that particular numbers are due (and should be selected) because they haven t been coming up often; others use the theory that some numbers are cold (and should be avoided) because they haven t been coming up often; and still others use astrology, numerology, or dreams. Because selections of winning lottery number combinations are independent events, such theories are worthless. A valid approach is to choose numbers that are rare in the sense that they are not selected by other people, so that if you win, you will not need to share your jackpot with many others. The combination of 1, 2, 3, 4, 5, 6 is a poor choice because many people tend to select it. In a Florida lottery with a $105 million prize, 52,000 tickets had 1, 2, 3, 4, 5, 6; if that combination had won, the prize would have been only $1000. It s wise to pick combinations not selected by many others. Avoid combinations that form a pattern on the entry card.
9 210 Chapter 5 Discrete Probability Distributions Formula 5-3 is a shortcut version that will always produce the same result as Formula 5-2. Although Formula 5-3 is usually easier to work with, Formula 5-2 is easier to understand directly. Based on Formula 5-2, we can express the standard deviation as s = 2 [(x - m) 2 # P (x)] or as the equivalent form given in Formula 5-4. When applying Formulas 5-1 through 5-4, use this rule for rounding results. Round-off Rule for M, S, and S 2 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 M, S, and S 2 to one decimal place. It is sometimes necessary to use a different rounding rule because of special circumstances, such as results that require more decimal places to be meaningful. For example, with four-engine jets the mean number of jet engines working successfully throughout a flight is , which becomes 4.0 when rounded to one more decimal place than the original data. Here, 4.0 would be misleading because it suggests that all jet engines always work successfully. We need more precision to correctly reflect the true mean, such as the precision in the number Identifying Unusual Results with the Range Rule of Thumb The range rule of thumb (introduced in Section 3-3) may be helpful in interpreting the value of a standard deviation. According to the range rule of thumb, most values should lie within 2 standard deviations of the mean; it is unusual for a value to differ from the mean by more than 2 standard deviations. (The use of 2 standard deviations is not an absolutely rigid value, and other values such as 3 could be used instead.) We can therefore identify unusual values by determining that they lie outside of these limits: Range Rule of Thumb maximum usual value M 2S minimum usual value M 2S CAUTION Know that the use of the number 2 in the range rule of thumb is somewhat arbitrary, and this rule is a guideline, not an absolutely rigid rule. Identifying Unusual Results with Probabilities Rare Event Rule for Inferential Statistics 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.
10 5-2 Random Variables 211 Probabilities can be used to apply the rare event rule as follows: Using Probabilities to Determine When Results Are Unusual Unusually high number of successes: x successes among n trials is an unusually high number of successes if the probability of x or more successes is unlikely with a probability of 0.05 or less. This criterion can be expressed as follows: P (x or more) 0.05.* Unusually low number of successes: x successes among n trials is an unusually low number of successes if the probability of x or fewer successes is unlikely with a probability of 0.05 or less. This criterion can be expressed as follows: P (x or fewer) 0.05.* *The value 0.05 is not absolutely rigid. Other values, such as 0.01, could be used to distinguish between results that can easily occur by chance and events that are very unlikely to occur by chance. Study hint: Take time to carefully read and understand the above rare event rule and the following paragraph. The next paragraph illustrates an extremely important approach used often in statistics. Suppose you were tossing a coin to determine whether it favors heads, and suppose 1000 tosses resulted in 501 heads. This is not evidence that the coin favors heads, because it is very easy to get a result like 501 heads in 1000 tosses just by chance. Yet, the probability of getting exactly 501 heads in 1000 tosses is actually quite small: This low probability reflects the fact that with 1000 tosses, any specific number of heads will have a very low probability. However, we do not consider 501 heads among 1000 tosses to be unusual, because the probability of 501 or more heads is high: Identifying Unusual Results with the Range Rule of Thumb In Example 5 we found that for groups of 5 offspring (generated from parents both having the green> yellow pair of genes), the mean number of peas with green pods is 3.8, and the standard deviation is 1.0. Use those results and the range rule of thumb to find the maximum and minimum usual values. Based on the results, determine whether it is unusual to generate 5 offspring peas and find that only 1 of them has a green pod. Using the range rule of thumb, we can find the maximum and minimum usual values as follows: maximum usual value: m + 2s = (1.0) = 5.8 minimum usual value: m - 2s = 3.8-2(1.0) = 1.8 Based on these results, we conclude that for groups of 5 offspring peas, the number of offspring peas with green pods should usually fall between 1.8 and 5.8. If 5 offspring peas are generated as described, it would be unusual to get only 1 with a green pod (because the value of 1 is outside of this range of usual values: 1.8 to 5.8). (In this case, the maximum usual value is actually 5, because that is the largest possible number of peas with green pods.)
11 212 Chapter 5 Discrete Probability Distributions 7 Identifying Unusual Results with Probabilities Use probabilities to determine whether 1 is an unusually low number of peas with green pods when 5 offspring are generated from parents both having the green> yellow pair of genes. To determine whether 1 is an unusually low number of peas with green pods (among 5 offspring), we need to find the probability of getting 1 or fewer peas with green pods. By referring to Table 5-1 on page 205 we can easily get the following results: P(1 or fewer) = P(1 or 0) = = Because the probability is less than 0.05, we conclude that the result of 1 pea with a green pod is unusually low. There is a very small likelihood (0.016) of getting 1 or fewer peas with green pods. Expected Value The mean of a discrete random variable is the theoretical mean outcome for infinitely many trials. We can think of that mean as the expected value in the sense that it is the average value that we would expect to get if the trials could continue indefinitely. The uses of expected value (also called expectation, or mathematical expectation) are extensive and varied, and they play an important role in decision theory. The expected value of a discrete random variable is denoted by E, and it represents the mean value of the outcomes. It is obtained by finding the value of [x # P (x)] E = [x # P(x)] CAUTION An expected value need not be a whole number, even if the different possible values of x might all be whole numbers. From Formula 5-1 we see that E = m. That is, the mean of a discrete random variable is the same as its expected value. For example, when generating groups of five offspring peas, the mean number of peas with green pods is 3.8 (see Table 5-3). So, it follows that the expected value of the number of peas with green pods is also 3.8. Because the concept of expected value is used often in decision theory, the following example involves a real decision. 8 How to Be a Better Bettor You are considering placing a bet either on the number 7 in roulette or on the pass line in the dice game of craps at the Venetian casino in Las Vegas. a. If you bet $5 on the number 7 in roulette, the probability of losing $5 is 37>38 and the probability of making a net gain of $175 is 1>38. (The prize is $180,
12 5-2 Random Variables 213 including your $5 bet, so the net gain is $175.) Find your expected value if you bet $5 on the number 7 in roulette. b. If you bet $5 on the pass line in the dice game of craps, the probability of losing $5 is 251> 495 and the probability of making a net gain of $5 is 244> 495. (If you bet $5 on the Pass Line and win, you are given $10 that includes your bet, so the net gain is $5.) Find your expected value if you bet $5 on the Pass Line. Which bet is better: A $5 bet on the number 7 in roulette or a $5 bet on the pass line in the dice game? Why? a. Roulette The probabilities and payoffs for betting $5 on the number 7 in roulette are summarized in Table 5-4. Table 5-4 also shows that the expected value is [x # P (x)] = -26. That is, for every $5 bet on the number 7, you can expect to lose an average of 26. Table 5-4 Roulette Event x P(x) x P(x) Lose -$5 37> 38 -$4.87 Gain (net) $175 1> 38 $4.61 Total -$0.26 # (or -26 ) b. Dice The probabilities and payoffs for betting $5 on the pass line in craps are summarized in Table 5-5. Table 5-5 also shows that the expected value is [x # P (x)] = -8. That is, for every $5 bet on the Pass Line, you can expect to lose an average of 8. Table 5-5 Dice Event x P(x) x P(x) Lose -$5 251> 495 -$2.54 Gain (net) $5 244> 495 $2.46 Total -$0.08 # (or -8 ) The $5 bet in roulette results in an expected value of -26 and the $5 bet in craps results in an expected value of -8. The bet in the dice game is better because it has the larger expected value. That is, you are better off losing 8 instead of losing 26. Even though the roulette game provides an opportunity for a larger payoff, the craps game is better in the long run. In this section we learned that a random variable has a numerical value associated with each outcome of some random procedure, and a probability distribution has a probability associated with each value of a random variable. We examined methods for finding the mean, variance, and standard deviation for a probability distribution. We saw that the expected value of a random variable is really the same as the mean. Finally, the range rule of thumb or probabilities can be used for determining when outcomes are unusual.
13 214 Chapter 5 Discrete Probability Distributions 5-2 Basic Skills and Concepts Statistical Literacy and Critical Thinking 1. Random Variable What is a random variable? A friend of the author buys one lottery ticket every week in one year. Over the 52 weeks, she counts the number of times that she won something. In this context, what is the random variable, and what are its possible values? 2. Expected Value A researcher calculates the expected value for the number of girls in three births. He gets a result of 1.5. He then rounds the result to 2, saying that it is not possible to get 1.5 girls when three babies are born. Is this reasoning correct? Explain. 3. Probability Distribution One of the requirements of a probability distribution is that the sum of the probabilities must be 1 (with a small discrepancy allowed for rounding errors). What is the justification for this requirement? 4. Probability Distribution A professional gambler claims that he has loaded a die so that the outcomes of 1, 2, 3, 4, 5, 6 have corresponding probabilities of 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6. Can he actually do what he has claimed? Is a probability distribution described by listing the outcomes along with their corresponding probabilities? Identifying Discrete and Continuous Random Variables. In Exercises 5 and 6, identify the given random variable as being discrete or continuous. 5. a. The number of people now driving a car in the United States b. The weight of the gold stored in Fort Knox c. The height of the last airplane that departed from JFK Airport in New York City d. The number of cars in San Francisco that crashed last year e. The time required to fly from Los Angeles to Shanghai 6. a. The total amount (in ounces) of soft drinks that you consumed in the past year b. The number of cans of soft drinks that you consumed in the past year c. The number of movies currently playing in U.S. theaters d. The running time of a randomly selected movie e. The cost of making a randomly selected movie Identifying Probability Distributions. In Exercises 7 12, determine whether or not a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied. 7. Genetic Disorder Three males with an X-linked genetic disorder have one child each. The random variable x is the number of children among the three who inherit the X-linked genetic disorder. 8. Caffeine Nation In the accompanying table, the random variable x represents the number of cups or cans of caffeinated beverages consumed by Americans each day (based on data from the National Sleep Foundation). 9. Overbooked Flights Air America has a policy of routinely overbooking flights. The random variable x represents the number of passengers who cannot be boarded because there are more passengers than seats (based on data from an IBM research paper by Lawrence, Hong, and Cherrier). x P (x) x P (x) x P (x)
14 5-2 Random Variables Eye Color Groups of five babies are randomly selected. In each group, the random variable x is the number of babies with green eyes (based on data from a study by Dr. Sorita Soni at Indiana University). (The symbol 0+ denotes a positive probability value that is very small.) 11. American Televisions In the accompanying table, the random variable x represents the number of televisions in a household in the United States (based on data from Frank N. Magid Associates). 12. TV Ratings In a study of television ratings, groups of 6 U.S. households are randomly selected. In the accompanying table, the random variable x represents the number of households among 6 that are tuned to 60 Minutes during the time that the show is broadcast (based on data from Nielsen Media Research). Pea Hybridization Experiment. In Exercises 13 16, refer to the accompanying table, which describes results from eight offspring peas. The random variable x represents the number of offspring peas with green pods. 13. Mean and Standard Deviation Find the mean and standard deviation for the numbers of peas with green pods. 14. Range Rule of Thumb for Unusual Events Use the range rule of thumb to identify a range of values containing the usual number of peas with green pods. Based on the result, is it unusual to get only one pea with a green pod? Explain. 15. Using Probabilities for Unusual Events a. Find the probability of getting exactly 7 peas with green pods. b. Find the probability of getting 7 or more peas with green pods. c. Which probability is relevant for determining whether 7 is an unusually high number of peas with green pods: the result from part (a) or part (b)? d. Is 7 an unusually high number of peas with green pods? Why or why not? 16. Using Probabilities for Unusual Events x P (x) x P (x) x P (x) Probabilities of Numbers of Peas with Green Pods Among 8 Offspring Peas x (Number of Peas with Green Pods) P (x) a. Find the probability of getting exactly 3 peas with green pods. b. Find the probability of getting 3 or fewer peas with green pods. c. Which probability is relevant for determining whether 3 is an unusually low number of peas with green pods: the result from part (a) or part (b)? d. Is 3 an unusually low number of peas with green pods? Why or why not? 17. Baseball World Series Based on past results found in the Information Please Almanac, there is a probability that a baseball World Series contest will last four games, a probability that it will last five games, a probability that it will last six games, and a probability that it will last seven games. a. Does the given information describe a probability distribution? b. Assuming that the given information describes a probability distribution, find the mean and standard deviation for the numbers of games in World Series contests. c. Is it unusual for a team to sweep by winning in four games? Why or why not?
15 216 Chapter 5 Discrete Probability Distributions 18. Job Interviews Based on information from MRINetwork, some job applicants are required to have several interviews before a decision is made. The number of required interviews and the corresponding probabilities are: 1 (0.09); 2 (0.31); 3 (0.37); 4 (0.12); 5 (0.05); 6 (0.05). a. Does the given information describe a probability distribution? b. Assuming that a probability distribution is described, find its mean and standard deviation. c. Use the range rule of thumb to identify the range of values for usual numbers of interviews. d. Is it unusual to have a decision after just one interview? Explain. 19. Bumper Stickers Based on data from CarMax.com, when a car is randomly selected, the number of bumper stickers and the corresponding probabilities are: 0 (0.824); 1 (0.083); 2 (0.039); 3 (0.014); 4 (0.012); 5 (0.008); 6 (0.008); 7 (0.004); 8 (0.004); 9 (0.004). a. Does the given information describe a probability distribution? b. Assuming that a probability distribution is described, find its mean and standard deviation. c. Use the range rule of thumb to identify the range of values for usual numbers of bumper stickers. d. Is it unusual for a car to have more than one bumper sticker? Explain. 20. Gender Discrimination The Telektronic Company hired 8 employees from a large pool of applicants with an equal number of males and females. If the hiring is done without regard to sex, the numbers of females hired and the corresponding probabilities are as follows: 0 (0.004); 1 (0.031); 2 (0.109); 3 (0.219); 4 (0.273); 5 (0.219); 6 (0.109); 7 (0.031); 8 (0.004). a. Does the given information describe a probability distribution? b. Assuming that a probability distribution is described, find its mean and standard deviation. c. Use the range rule of thumb to identify the range of values for usual numbers of females hired in such groups of eight. d. If the most recent group of eight newly hired employees does not include any females, does there appear to be discrimination based on sex? Explain. 21. Finding Mean and Standard Deviation Let the random variable x represent the number of girls in a family of three children. Construct a table describing the probability distribution, then find the mean and standard deviation. (Hint: List the different possible outcomes.) Is it unusual for a family of three children to consist of three girls? 22. Finding Mean and Standard Deviation Let the random variable x represent the number of girls in a family of four children. Construct a table describing the probability distribution, then find the mean and standard deviation. (Hint: List the different possible outcomes.) Is it unusual for a family of four children to consist of four girls? 23. Random Generation of Telephone Numbers A description of a Pew Research Center poll referred to the random generation of the last two digits of telephone numbers. Each digit has the same chance of being randomly generated. Construct a table representing the probability distribution for digits randomly generated by computer, find its mean and standard deviation, then describe the shape of the probability histogram. 24. Analysis of Leading Digits The analysis of the leading (first) digits of checks led to the conclusion that companies in Brooklyn, New York, were guilty of fraud. For the purposes of this exercise, assume that the leading digits of check amounts are randomly generated by computer. a. Identify the possible leading digits. b. Find the mean and standard deviation of such leading digits. c. Use the range rule of thumb to identify the range of usual values. d. Can any leading digit be considered unusual? Why or why not?
16 5-2 Random Variables Finding Expected Value for the Illinois Pick 3 Game In the Illinois Pick 3 lottery game, you pay 50 to select a sequence of three digits, such as 233. If you select the same sequence of three digits that are drawn, you win and collect $250. a. How many different selections are possible? b. What is the probability of winning? c. If you win, what is your net profit? d. Find the expected value. e. If you bet 50 in Illinois Pick 4 game, the expected value is -25. Which bet is better: A 50 bet in the Illinois Pick 3 game or a 50 bet in the Illinois Pick 4 game? Explain. 26. Expected Value in New Jersey s Pick 4 Game In New Jersey s Pick 4 lottery game, you pay 50 to select a sequence of four digits, such as If you select the same sequence of four digits that are drawn, you win and collect $2788. a. How many different selections are possible? b. What is the probability of winning? c. If you win, what is your net profit? d. Find the expected value. e. If you bet 50 in Illinois Pick 4 game, the expected value is -25. Which bet is better: A 50 bet in the Illinois Pick 4 game or a 50 bet in New Jersey s Pick 4 game? Explain. 27. Expected Value in Roulette When playing roulette at the Bellagio casino in Las Vegas, a gambler is trying to decide whether to bet $5 on the number 13 or to bet $5 that the outcome is any one of these five possibilities: 0 or 00 or 1 or 2 or 3. From Example 8, we know that the expected value of the $5 bet for a single number is -26. For the $5 bet that the outcome is 0 or 00 or 1 or 2 or 3, there is a probability of 5> 38 of making a net profit of $30 and a 33> 38 probability of losing $5. a. Find the expected value for the $5 bet that the outcome is 0 or 00 or 1 or 2 or 3. b. Which bet is better: A $5 bet on the number 13 or a $5 bet that the outcome is 0 or 00 or 1 or 2 or 3? Why? 28. Expected Value for Deal or No Deal The television game show Deal or No Deal begins with individual suitcases containing the amounts of 1, $1, $5, $10, $25, $50, $75, $100, $200, $300, $400, $500, $750, $1000, $5000, $10,000, $25,000, $50,000, $75,000, $100,000, $200,000, $300,000, $400,000, $500,000, $750,000, and $1,000,000. If a player adopts the strategy of choosing the option of no deal until one suitcase remains, the payoff is one of the amounts listed, and they are all equally likely. a. Find the expected value for this strategy. b. Find the value of the standard deviation. c. Use the range rule of thumb to identify the range of usual outcomes. d. Based on the preceding results, is a result of $750,000 or $1,000,000 unusual? Why or why not? 29. Expected Value for Life Insurance There is a probability that a randomly selected 30-year-old male lives through the year (based on data from the U.S. Department of Health and Human Services). A Fidelity life insurance company charges $161 for insuring that the male will live through the year. If the male does not survive the year, the policy pays out $100,000 as a death benefit. a. From the perspective of the 30-year-old male, what are the values corresponding to the two events of surviving the year and not surviving? b. If a 30-year-old male purchases the policy, what is his expected value? c. Can the insurance company expect to make a profit from many such policies? Why? 30. Expected Value for Life Insurance There is a probability that a randomly selected 50-year-old female lives through the year (based on data from the U.S. Department of Health and Human Services). A Fidelity life insurance company charges $226 for insuring
17 218 Chapter 5 Discrete Probability Distributions that the female will live through the year. If she does not survive the year, the policy pays out $50,000 as a death benefit. a. From the perspective of the 50-year-old female, what are the values corresponding to the two events of surviving the year and not surviving? b. If a 50-year-old female purchases the policy, what is her expected value? c. Can the insurance company expect to make a profit from many such policies? Why? 5-2 Beyond the Basics 31. Junk Bonds Kim Hunter has $1000 to invest, and her financial analyst recommends two types of junk bonds. The A bonds have a 6% annual yield with a default rate of 1%. The B bonds have an 8% annual yield with a default rate of 5%. (If the bond defaults, the $1000 is lost.) Which of the two bonds is better? Why? Should she select either bond? Why or why not? 32. Defective Parts: Finding Mean and Standard Deviation The Sky Ranch is a supplier of aircraft parts. Included in stock are eight altimeters that are correctly calibrated and two that are not. Three altimeters are randomly selected without replacement. Let the random variable x represent the number that are not correctly calibrated. Find the mean and standard deviation for the random variable x. 33. Labeling Dice to Get a Uniform Distribution Assume that you have two blank dice, so that you can label the 12 faces with any numbers. Describe how the dice can be labeled so that, when the two dice are rolled, the totals of the two dice are uniformly distributed in such a way that the outcomes of 1, 2, 3, Á, 12 each have probability 1>12. (See Can One Load a Set of Dice So That the Sum Is Uniformly Distributed? by Chen, Rao, and Shreve, Mathematics Magazine, Vol. 70, No. 3.) 5-3 Binomial Probability Distributions Key Concept In this section we focus on one particular category of discrete probability distributions: binomial probability distributions. Because binomial probability distributions involve proportions used with methods of inferential statistics discussed later in this book, it is important to understand fundamental properties of this particular class of probability distributions. In this section we present a basic definition of a binomial probability distribution along with notation, and methods for finding probability values. As in other sections, we want to interpret probability values to determine whether events are usual or unusual. Binomial probability distributions allow us to deal with circumstances in which the outcomes belong to two relevant categories, such as acceptable>defective or survived>died. Other requirements are given in the following definition. A binomial probability distribution results from a procedure that meets all the following requirements: 1. The procedure has a fixed number of trials. 2. The trials must be independent. (The outcome of any individual trial doesn t affect the probabilities in the other trials.)
18 5-3 Binomial Probability Distributions Each trial must have all outcomes classified into two categories (commonly referred to as success and failure). 4. The probability of a success remains the same in all trials. Independence Requirement When selecting a sample (such as survey subjects) for some statistical analysis, we usually sample without replacement. Recall that sampling without replacement involves dependent events, which violates the second requirement in the above definition. However, we can often assume independence by applying the following 5% guideline introduced in Section 4-4: Treating Dependent Events as Independent: The 5% Guideline for Cumbersome Calculations If calculations are cumbersome and if a sample size is no more than 5% of the size of the population, treat the selections as being independent (even if the selections are made without replacement, so that they are technically dependent). If a procedure satisfies the above four requirements, the distribution of the random variable x (number of successes) is called a binomial probability distribution (or binomial distribution). The following notation is commonly used. Notation for Binomial Probability Distributions S and F (success and failure) denote the two possible categories of all outcomes. P(S) = p P (F) = 1 - p = q n x p q P (x) ( p = probability of a success) (q = probability of a failure) denotes the fixed number of trials. denotes a specific number of successes in n trials, so x can be any whole number between 0 and n, inclusive. denotes the probability of success in one of the n trials. denotes the probability of failure in one of the n trials. denotes the probability of getting exactly x successes among the n trials. Not At Home Pollsters cannot simply ignore those who were not at home when they were called the first time. One solution is to make repeated callback attempts until the person can be reached. Alfred Politz and Willard Simmons describe a way to compensate for those missing results without making repeated callbacks. They suggest weighting results based on how often people are not at home. For example, a person at home only two days out of six will have a 2>6 or 1>3 probability of being at home when called the first time. When such a person is reached the first time, his or her results are weighted to count three times as much as someone who is always home. This weighting is a compensation for the other similar people who are home two days out of six and were not at home when called the first time. This clever solution was first presented in The word success as used here is arbitrary and does not necessarily represent something good. Either of the two possible categories may be called the success S as long as its probability is identified as p. (The value of q can always be found by subtracting p from 1; if p = 0.95, then q = = 0.05.) CAUTION When using a binomial probability distribution, always be sure that x and p both refer to the same category being called a success.
19 220 Chapter 5 Discrete Probability Distributions 1 Genetics Consider an experiment in which 5 offspring peas are generated from 2 parents each having the green> yellow combination of genes for pod color. Recall from the Chapter Problem that the probability an offspring pea will have a green pod is 3 4 or That is, P (green pod) = Suppose we want to find the probability that exactly 3 of the 5 offspring peas have a green pod. a. Does this procedure result in a binomial distribution? b. If this procedure does result in a binomial distribution, identify the values of n, x, p, and q. a. This procedure does satisfy the requirements for a binomial distribution, as shown below. 1. The number of trials (5) is fixed. 2. The 5 trials are independent, because the probability of any offspring pea having a green pod is not affected by the outcome of any other offspring pea. 3. Each of the 5 trials has two categories of outcomes: The pea has a green pod or it does not. 4. For each offspring pea, the probability that it has a green pod is 3> 4 or 0.75, and that probability remains the same for each of the 5 peas. b. Having concluded that the given procedure does result in a binomial distribution, we now proceed to identify the values of n, x, p, and q. 1. With 5 offspring peas, we have n = We want the probability of exactly 3 peas with green pods, so x = The probability of success (getting a pea with a green pod) for one selection is 0.75, so p = The probability of failure (not getting a green pod) is 0.25, so q = Again, it is very important to be sure that x and p both refer to the same concept of success. In this example, we use x to count the number of peas with green pods, so p must be the probability that a pea has a green pod. Therefore, x and p do use the same concept of success (green pod) here. We now discuss three methods for finding the probabilities corresponding to the random variable x in a binomial distribution. The first method involves calculations using the binomial probability formula and is the basis for the other two methods. The second method involves the use of computer software or a calculator, and the third method involves the use of Table A-1. (With technology so widespread, such tables are becoming obsolete.) If you are using computer software or a calculator that automatically produces binomial probabilities, we recommend that you solve one or two exercises using Method 1 to ensure that you understand the basis for the calculations. Understanding is always infinitely better than blind application of formulas.