7 CHAPTER The Binomial Distribution Introduction Many probability problems involve assigning probabilities to the outcomes of a probability experiment. These probabilities and the corresponding outcomes make up a probability distribution. There are many different probability distributions. One special probability distribution is called the binomial distribution. The binomial distribution has many uses such as in gambling, in inspecting parts, and in other areas. 114 Copyright © 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use. Discrete Probability Distributions In mathematics, a variable can assume different values. For example, if one records the temperature outside every hour for a 24-hour period, temperature is considered a variable since it assumes different values. Variables whose values are due to chance are called random variables. When a die is rolled, the value of the spots on the face up occurs by chance; hence, the number of spots on the face up on the die is considered to be a random variable. The outcomes of a die are 1, 2, 3, 4, 5, and 6, and the probability of each outcome occurring is 1 6 . The outcomes and their corresponding probabilities can be written in a table, as shown, and make up what is called a probability distribution. Value, x 123456 Probability, P(x) 1 6 1 6 1 6 1 6 1 6 1 6 A probability distribution consists of the values of a random variable and their corresponding probabilities. There are two kinds of probability distributions. They are discrete and continuous.Adiscrete variable has a countable number of values (countable means values of zero, one, two, three, etc.). For example, when four coins are tossed, the outcomes for the number of heads obtained are zero, one, two, three, and four. When a single die is rolled, the outcomes are one, two, three, four, five, and six. These are examples of discrete variables. A continuous variable has an infinite number of values between any two values. Continuous variables are measured. For example, temperature is a continuous variable since the variable can assume any value between 108 and 208 or any other two temperatures or values for that matter. Height and weight are continuous variables. Of course, we are limited by our measuring devices and values of continuous variables are usually ‘‘rounded off.’’ EXAMPLE: Construct a discrete probability distribution for the number of heads when three coins are tossed. SOLUTION: Recall that the sample space for tossing three coins is TTT, TTH, THT, HTT, HHT, HTH, THH, and HHH. CHAPTER 7 The Binomial Distribution 115 The outcomes can be arranged according to the number of heads, as shown. 0 heads TTT 1 head TTH, THT, HTT 2 heads THH, HTH, HHT 3 heads HHH Finally, the outcomes and corresponding probabilities can be written in a table, as shown. Outcome, x 0123 Probability, P(x) 1 8 3 8 3 8 1 8 The sum of the probabilities of a probability distribution must be 1. A discrete probability distribution can also be shown graphically by labeling the x axis with the values of the outcomes and letting the values on the y axis represent the probabilities for the outcomes. The graph for the discrete probability distribution of the number of heads occurring when three coins are tossed is shown in Figure 7-1. There are many kinds of discrete probability distributions; however, the distribution of the number of heads when three coins are tossed is a special kind of distribution called a binomial distribution. Fig. 7-1. CHAPTER 7 The Binomial Distribution 116 A binomial distribution is obtained from a probability experiment called a binomial experiment. The experiment must satisfy these conditions: 1. Each trial can have only two outcomes or outcomes that can be reduced to two outcomes. The outcomes are usually considered as a success or a failure. 2. There is a fixed number of trials. 3. The outcomes of each trial are independent of each other. 4. The probability of a success must remain the same for each trial. EXAMPLE: Explain why the probability experiment of tossing three coins is a binomial experiment. SOLUTION: In order to be a binomial experiment, the probability experiment must satisfy the four conditions explained previously. 1. There are only two outcomes for each trial, head and tail. Depending on the situation, either heads or tails can be defined as a success and the other as a failure. 2. There is a fixed number of trials. In this case, there are three trials since three coins are tossed or one coin is tossed three times. 3. The outcomes are independent since tossing one coin does not effect the outcome of the other two tosses. 4. The probability of a success (say heads) is 1 2 and it does not change. Hence the experiment meets the conditions of a binomial experiment. Now consider rolling a die. Since there are six outcomes, it cannot be considered a binomial experiment. However, it can be made into a binomial experiment by considering the outcome of getting five spots (for example) a success and every other outcome a failure. In order to determine the probability of a success for a single trial of a probability experiment, the following formula can be used. n C x ÁðpÞ x Áð1 À pÞ nÀx where n ¼ the total number of trials x ¼ the number of successes (1, 2, 3, ., n) p ¼ the probability of a success The formula has three parts: n C x determines the number of ways a success can occur. ( p) x is the probability of getting x successes, and (1 À p) nÀx is the probability of getting n À x failures. CHAPTER 7 The Binomial Distribution 117 EXAMPLE: A coin is tossed 3 times. Find the probability of getting two heads and a tail in any given order. SOLUTION: Since the coin is tossed 3 times, n ¼ 3. The probability of getting a head (suc- cess) is 1 2 ,sop ¼ 1 2 and the probability of getting a tail (failure) is 1 À 1 2 ¼ 1 2 ; x ¼ 2 since the problem asks for 2 heads. (n À x) ¼ 3 À 2 ¼ 1. Hence, Pð2 headsÞ¼ 3 C 2 Á 1 2  2 1 2  1 ¼ 3 Á 1 4  1 2  ¼ 3 8 Notice that there were 3 C 2 or 3 ways to get two heads and a tail. The answer 3 8 is also the same as the answer obtained using classical probability that was shown in the first example in this chapter. EXAMPLE: A die is rolled 3 times; find the probability of getting exactly one five. SOLUTION: Since we are rolling a die 3 times, n ¼ 3. The probability of getting a 5 is 1 6 . The probability of not getting a 5 is 1 À 1 6 or 5 6 . Since a success is getting one five, x ¼ 1 and n À x ¼ 3 À 1 ¼ 2. Hence, Pðone 5Þ¼ 3 C 1 Á 1 6  1 Á 5 6  2 ¼ 3 Á 1 6 Á 25 36 ¼ 25 72 or 0:3472 About 35% of the time, exactly one 5 will occur. CHAPTER 7 The Binomial Distribution 118 EXAMPLE: An archer hits the bull’s eye 80% of the time. If he shoots 5 arrows, find the probability that he will get 4 bull’s eyes. SOLUTION: n ¼ 5, x ¼ 4, p ¼ 0.8, 1 À p ¼ 1 À 0.8 ¼ 0.2 Pð4 bull’s eyesÞ¼ 5 C 4 ð0:8Þ 4 ð0:2Þ 1 ¼ 5 Á 0:08192 ¼ 0:4096 In order to construct a probability distribution, the following formula is used: n C x p x (1 À p) n À x where x ¼ 1, 2, 3, .n. The next example shows how to use the formula. EXAMPLE: A die is rolled 3 times. Construct a probability distribution for the number of fives that will occur. SOLUTION: In this case, the die is tossed 3 times, so n ¼ 3. The probability of getting a 5 on a die is 1 6 , and one can get x ¼ 0, 1, 2, or 3 fives. For x ¼ 0, 3 C 0 1 6  0 5 6  3 ¼ 0:5787 For x ¼ 1, 3 C 1 1 6  1 5 6  2 ¼ 0:3472 For x ¼ 2, 3 C 2 1 6  2 5 6  1 ¼ 0:0694 For x ¼ 3, 3 C 3 1 6  3 5 6  0 ¼ 0:0046 Hence, the probability distribution is Number of fives, x 0123 Probability, P(x) 0.5787 0.3472 0.0694 0.0046 CHAPTER 7 The Binomial Distribution 119 Note: Most statistics books have tables that can be used to compute probabilities for binomial variables. PRACTICE 1. A student takes a 5-question true–false quiz. Since the student has not studied, he decides to flip a coin to determine the answers. What is the probability that the student guesses exactly 3 out of 5 correctly? 2. A basketball player makes three-fourths of his free throws. Assume each shot is independent of another shot. Find the probability that he makes the next four free throws. 3. A circuit has 6 breakers. The probability that each breaker will fail is 0.1. If the circuit is activated, find the probability that exactly two breakers will fail. Each breaker is independent of any other breaker. 4. Eight coins are tossed; find the probability of getting exactly 3 heads. 5. A box contains 4 red marbles and 2 white marbles. A marble is drawn and replaced four times. Find the probability of getting exactly 3 red marbles and one white marble. ANSWERS 1. n ¼ 5, x ¼ 3, p ¼ 1 2 P(exactly 3 correct) ¼ 5 C 3 1 2  3 1 2  2 ¼ð10Þ 1 32 ¼ 5 16 ¼ 0:3125 2. n ¼ 4, x ¼ 4, p ¼ 3 4 Pð4 successesÞ¼ 4 C 4 3 4  4 1 4  0 ¼ 81 256 % 0:3164 3. n ¼ 6, x ¼ 2, p ¼ (0.1) Pð2 will failÞ¼ 6 C 2 ð0:1Þ 2 ð0:9Þ 4 ¼ 15ð0:006561Þ¼0:098415 CHAPTER 7 The Binomial Distribution 120 4. n ¼ 8, x ¼ 3, p ¼ 1 2 Pð3 headsÞ¼ 8 C 3 1 2  3 1 2  5 ¼ 56 1 8  1 32  ¼ 56 Á 1 256 ¼ 7 32 ¼ 0:21875 5. n ¼ 4, x ¼ 3, p ¼ 2 3 Pð3 red marblesÞ¼ 4 C 3 2 3  3 1 3  1 ¼ 4 Á 8 81 ¼ 32 81 % 0:395 The Mean and Standard Deviation for a Binomial Distribution Suppose you roll a die many times and record the number of threes you obtain. Is it possible to predict ahead of time the average number of threes you will obtain? The answer is ‘‘Yes.’’ It is called expected value or the mean of a binomial distribution. This mean can be found by using the formula mean (") ¼ np where n is the number of times the experiment is repeated and p is the probability of a success. The symbol for the mean is the Greek letter " (mu). EXAMPLE: A die is tossed 180 times and the number of threes obtained is recorded. Find the mean or expected number of threes. CHAPTER 7 The Binomial Distribution 121 SOLUTION: n ¼ 180 and p ¼ 1 6 since there is one chance in 6 to get a three on each roll. " ¼ n Á p ¼ 180 Á 1 6 ¼ 30 Hence, one would expect on average 30 threes. EXAMPLE: Twelve cards are selected from a deck and each card is replaced before the next one is drawn. Find the average number of diamonds. SOLUTION: In this case, n ¼ 12 and p ¼ 13 52 or 1 4 since there are 13 diamonds and a total of 52 cards. The mean is " ¼ n Á p ¼ 12 Á 1 4 ¼ 3 Hence, on average, we would expect 3 diamonds in the 12 draws. Statisticians are not only interested in the average of the outcomes of a probability experiment but also in how the results of a probability experiment vary from trial to trial. Suppose we roll a die 180 times and record the number of threes obtained. We know that we would expect to get about 30 threes. Now what if the experiment was repeated again and again? In this case, the number of threes obtained each time would not always be 30 but would vary about the mean of 30. For example, we might get 28 threes one time and 34 threes the next time, etc. How can this variability be explained? Statisticians use a measure called the standard deviation. When the standard deviation of a variable is large, the individual values of the variable are spread out from the mean of the distribution. When the standard deviation of a variable is small, the individual values of the variable are close to the mean. The formula for the standard deviation for a binomial distribution is standard deviation ' ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi npð1 À pÞ p . The symbol for the standard deviation is the Greek letter ' (sigma). CHAPTER 7 The Binomial Distribution 122 EXAMPLE: A die is rolled 180 times. Find the standard deviation of the number of threes. SOLUTION: n¼ 180, p¼ 1 6 ,1À p ¼ 1 À 1 6 ¼ 5 6 ' ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi npð1 À pÞ p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 180 Á 1 6 Á 5 6 r ¼ ffiffiffiffiffi 25 p ¼ 5 The standard deviation is 5. Now what does this tell us? Roughly speaking, most of the values fall within two standard deviations of the mean. " À 2'<most values <"þ 2' In the die example, we can expect most values will fall between 30 À 2 Á 5 < most values < 30 þ 2 Á 5 30 À 10 < most values < 30 þ 10 20 < most values < 40 In this case, if we did the experiment many times we would expect between 20 and 40 threes most of the time. This is an approximate ‘‘range of values.’’ Suppose we rolled a die 180 times and we got only 5 threes, what can be said? It can be said that this is an unusually small number of threes. It can happen by chance, but not very often. We might want to consider some other possibilities. Perhaps the die is loaded or perhaps the die has been manipulated by the person rolling it! EXAMPLE: An archer hits the bull’s eye 80% of the time. If he shoots 100 arrows, find the mean and standard deviation of the number of bull’s eyes. If he travels to many tournaments, find the approximate range of values. SOLUTION: n¼ 100, p¼ 0.80, 1À p¼ 1¼ 0.80¼ 0.20 CHAPTER 7 The Binomial Distribution 123 [...]... the right of it and the number above and to the left of it For example, the number 10 in the fifth row is found by adding the 4 and 6 in the fourth row The number 15 in the sixth row is found by adding the 5 and 10 in the previous row The numbers in each row represent the number of different outcomes when coins are tossed For example, the numbers in row 3 are 1, 3, 3, and 1 When 3 coins are tossed, the. .. many outcomes are there for a binomial experiment? a 0 b 1 c 2 d It varies CHAPTER 7 The Binomial Distribution 2 The sum of the probabilities of all outcomes in a probability distribution is a b c d 0 1 2 It varies 3 Which one is not a requirement of a binomial experiment? a b c d There are 2 outcomes for each trial There is a fixed number of trials The outcomes must be dependent The probability of... HTH THT HHH HHT TTH TTT Notice that there is only one way to get 3 heads There are 3 different ways to get 2 heads and a tail There are 3 different ways to get two tails and a head and there is one way to get 3 tails The same results apply to the genders of the children in a family with three children Another property of the triangle is that it represents the answer to the number of combination of n items... must be the same for all trials 4 The formula for the mean of a binomial distribution is a np b np(1 À p) c n(1 À p) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d npð1 À pÞ 5 The formula for a standard deviation of a binomial distribution is a np b np(1 À p) c n(1 À p) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d npð1 À pÞ 6 If 30% of commuters ride to work on a bus, find the probability that if 8 workers are selected at random, 3 will ride the bus... contributions to mathematics in areas of number theory, geometry, and probability He is credited along with Fermat for the beginnings of the formal study of probability He is given credit for developing a triangular array of numbers known as Pascal’s triangle, shown here CHAPTER 7 The Binomial Distribution 1 1 1 1 1 1 3 4 5 6 2 1 3 1 6 10 15 4 10 20 1 5 15 1 6 1 Each number in the triangle is the sum of the number... 130 CHAPTER 7 The Binomial Distribution The numbers in the triangle have applications in other areas of mathematics such as algebra and graph theory It is interesting to note that Pascal included his triangle in a book he wrote in 1653 It wasn’t printed until 1665 It is not known if Pascal developed the triangle on his own or heard about it from someone else; however, a similar version of the triangle... is taken, the standard deviation of the sample is a b c d 10 8.42 9.49 5 10 A survey found that 50% of adults get the daily news from radio If a sample of 64 adults is selected, the approximate range of the number of people who get their news from the radio is a b c d 24 30 28 26 and and and and 40 34 32 36 Probability Sidelight PASCAL’S TRIANGLE Blaise Pascal (1623–1662) was a French mathematician... d 0.361 0.482 0.254 0.323 7 If 10% of the people who take a certain medication get a headache, find the probability that if 5 people take the medication, one will get a headache a b c d 0.328 0.136 0.472 0.215 127 CHAPTER 7 The Binomial Distribution 128 8 A survey found that 30% of all Americans have eaten pizza for breakfast If 500 people are selected at random, the mean number of people who have eaten... on the standard deviation will be presented in Chapter 9 More information on the standard deviation can also be found in all statistics textbooks PRACTICE 1 Twenty cards are selected from a deck of 52 cards Each card is replaced before the next card is selected Find the mean and standard deviation of the number of clubs selected 2 A coin is tossed 1000 times Find the mean and standard deviation of the. .. choice exam is given There are four choices for each question Find the mean and standard deviation of the number of correct answers a student will get if he selects each answer at random 4 A die is rolled 720 times Find the mean, standard deviation, and approximate range of values for the number of threes obtained 5 A factory manufactures microchips of which 4% are defective Find the average number of . and letting the values on the y axis represent the probabilities for the outcomes. The graph for the discrete probability distribution of the number of. tossed is a special kind of distribution called a binomial distribution. Fig. 7-1. CHAPTER 7 The Binomial Distribution 116 A binomial distribution is obtained