Name Class Date Binomial Distributions Extension: Distributions Essential question: What is a probability distribution and how is it displayed? 8-6 CC.9 2.S.MD.5(+) ENGAGE Introducing Distributions Video Tutor A random variable is a variable whose value is determined by the outcome of a probability experiment. For example, when you roll a number cube, you can use a random variable X to represent the number you roll. The possible values of X are, 2, 3, 4, 5, and 6. A probability distribution is a data distribution that gives the probabilities of the values of a random variable. A probability distribution can be represented by a histogram in which the values of the random variable that is, the possible outcomes are on the horizontal axis, and probabilities are on the vertical axis. The figure shows the probability distribution for rolling a number cube. 3 6 2 6 6 When the values of a random variable are consecutive whole numbers, as is the case for rolling a number cube, a histogram for the probability distribution typically shows bars that each have a width of and is centered on a value of the variable. The area of each bar therefore equals the probability of the corresponding outcome, and the combined areas of the bars is the sum of the probabilities, which is. 2 3 4 5 6 Result of rolling number cube A cumulative probability is the probability that a random variable is less than or equal to a given value. You can find cumulative probabilities from a histogram by adding the areas of the bars for all outcomes less than or equal to the given value. REFLECT a. In an experiment in which a coin is tossed twice, the random variable X is the number of times that the coin lands heads up. What are the possible values of the random variable? b. A spinner has 8 equal sections, each labeled, 2, 3, or 4. The histogram shows the probability distribution for spinning the spinner. How many sections of the spinner are labeled with each number? How do you know? 2 3 8 4 8 2 3 4 Result of spinning a spinner Chapter 8 473 Lesson 6
2 CC.9 2.S.MD.5(+) EXAMPLE Displaying a Distribution You roll two number cubes at the same time. Let X be a random variable that represents the sum of the numbers rolled. Make a histogram to show the probability distribution for X. A Complete the frequency table to show the number of ways that you can get each sum in one roll of the number cubes. Sum 2 3 4 5 6 7 8 9 0 2 Frequency B Add the frequencies you found in part A to find the total number of possible outcomes. The total number of possible outcomes is. C Divide each frequency by the total number of outcomes to find the probability of each sum. Complete the table. Sum 2 3 4 5 6 7 8 9 0 2 36 D Create a histogram with the sums on the horizontal axis and the probabilities on the vertical axis. Complete the histogram below by labeling the axes and drawing a bar to represent the probability of each sum. 36 Sum REFLECT 2a. The probability that you roll a sum less than or equal to 5 is written P(X 5). What is this probability? How is it represented in the histogram? Chapter 8 474 Lesson 6
In the example, you used theoretical probabilities to define a probability distribution. You can also use experimental probabilities to define a probability distribution. 3 CC.9 2.S.IC.2 EXPLORE Using a Simulation You flip a coin 7 times in a row. Use a simulation to determine the probability distribution for the number of times the coin lands heads up. A When you flip a coin, the possible outcomes are heads and tails. You will use your calculator to generate random numbers between 0 and, assigning heads to numbers less than or equal to 0.5 and tails to numbers greater than 0.5. To do the simulation, press MATH and then select PRB. Choose :rand and press ENTER. Now press ENTER 7 times to generate 7 random numbers. This simulates one trial (that is, one set of 7 coin flips). Record the number of heads in the table. For example, on the calculator screen shown here, there are 3 numbers less than or equal to 0.5, so there are 3 heads. Carry out three more trials and record your results in the table. Trial 2 3 4 Number of Heads B Report your results to your teacher in order to combine everyone s results. Use the combined class data to complete the table below. To find the relative frequency for an outcome, divide the frequency of the outcome by the total number of trials in the class. Number of Heads 0 2 3 4 5 6 7 Frequency Relative Frequency C Enter the outcomes (0 through 7) into your calculator as list L. Enter the relative frequencies as list L 2. D Make a histogram by turning on a statistics plot, selecting the histogram option, and using L for Xlist and L 2 for Freq. Set the viewing window as shown. Then press GRAPH. A sample histogram is shown below. Chapter 8 475 Lesson 6
REFLECT 3a. Describe the shape of the probability distribution. 3b. Based on the histogram, what is P(X 3)? That is, what is the probability of getting 3 or fewer heads when you flip a coin 7 times? Explain. 3c. If you flipped a coin 7 times and got 7 heads, would this cause you to question whether the coin is fair? Why or why not? 4 CC.9 2.S.MD.3(+) example Analyzing a Distribution The histogram shows the theoretical probability distribution for the situation in the Explore. Use the distribution to answer each question. 0.3 0.273 0.273 A What is the probability of getting 4 or more heads? P(X 4) = P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) So, the probability of getting 4 or more heads is 0.2 0. 0.64 0.64 + + + =. B What is the probability of getting at least head? An easy way to calculate this probability is to use the complement of the event. The complement of getting at least head is getting 0 heads. Use the histogram to find P(X = 0) and subtract it from. P(X = 0) = So, the probability of getting at least head is - =. 0.055 0.055 0.008 0.008 0 2 3 4 5 6 7 Number of Heads REFLECT 4a. Why are the probabilities in the histogram you made in the Explore different from the probabilities given in the histogram above? Chapter 8 476 Lesson 6
4b. What do you think would happen to the histogram you made in the Explore if you included data from 000 additional trials? 4c. Why does it make sense that the histogram that shows the theoretical probabilities is symmetric? practice. The spinner at right has three equal sections. You spin the spinner twice and find the sum of the two numbers the spinner lands on. a. Let X be a random variable that represents the sum of the two numbers. What are the possible values of X? 2 3 b. Complete the table. Sum c. Make a histogram of the probability distribution. d. What is the probability that the sum is not 2? How is this probability represented in the histogram? Chapter 8 477 Lesson 6
2. You roll two number cubes at the same time. Let X be a random variable that represents the absolute value of the difference of the numbers rolled. a. What are the possible values of X? b. Complete the table. Difference c. Is this probability distribution symmetric? Why or why not? A trick coin is designed to land heads up with a probability of 80%. You flip the coin 7 times. The histogram shows the probability distribution for the number of times the coin lands heads up. ( 0+ means slightly greater than 0.) Use the histogram for Exercises 3-6. 3. What is the probability of getting 6 or 7 heads? 4. What is the probability of getting 4 or more heads? Explain. 0.4 0.3 0.2 0. 0.367 0.275 0.20 0.029 0.004 0.5 0+ 0+ 0 2 3 4 5 6 7 Number of Heads 5. Is the probability of getting an even number of heads the same as the probability of getting an odd number of heads? Explain. 6. Suppose you flip a coin 7 times and get 7 heads. Based on what you know now, would you question whether the coin is fair? Why or why not? Chapter 8 478 Lesson 6
Name Class Date 8-6 Additional Practice. You roll two four-sided number cubes at the same time. Let X be a random variable that represents the product of the numbers rolled. a. What are the possible values of X? b. Complete the table. Control c. Draw a histogram of the probability distribution. 2. Describe the probability distribution for flipping a fair coin. Does this depend on the number of flips? Chapter 8 479 Lesson 6
Problem Solving Sales records for the snack machines show that of every 6 students buys a bag of trail mix. There are 5 students waiting to use the machines. Melanie uses the formula for binomial probability, P(r) = n C r p r q n r, to determine the number of students expected to buy n! ). r! n r! trail mix. (The expression n C r means ( )( ). What is the probability of exactly 3 students buying a bag of trail mix? a. What is the probability of each student buying a bag of trail mix? b. Define each variable used in the formula and give its value. c. Write the binomial formula, substituting these values. d. Solve the equation to give the probability of exactly 3 students buying a bag of trail mix. 2. Repeat the process to find the probability of exactly 0,, 2, 4, and 5 students buying a bag of trail mix. Use these results to graph a probability distribution. 3. What is the probability of at least student buying a bag of trail mix? a. Describe a method to solve involving the sum of probabilities. b. Describe a method to solve that uses the formula P(E) +P(not E) =. c. Use either method to determine the probability of at least student buying a bag of trail mix. Chapter 8 480 Lesson 6