23.1 Probability Distributions

Transcription

1 3.1 Probability Distributions Essential Question: What is a probability distribution for a discrete random variable, and how can it be displayed? Explore Using Simulation to Obtain an Empirical Probability Distribution Resource Locker 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 the random variable X to represent the number you roll. The possible values of X are 1,, 3, 4, 5, and. 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 probability distribution for rolling a number cube is shown. Notice that it is a uniform distribution. When the values of a random variable are discrete, as is the case for rolling a number cube, a histogram for the probability distribution typically shows bars that each have a width of 1 and are 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 are the sum of the probabilities, which is 1. Probability Result of rolling a 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. Suppose you flip a coin 5 times in a row. Use a simulation to determine the probability distribution for the number of times the coin lands heads up. When you flip a coin, the possible outcomes are heads and tails. Use a graphing calculator to generate the integers 0 and 1 randomly, associating each 0 with tails and each 1 with heads. To do the simulation, press MATH and then select the probability (PRB) menu. Choose 5:randInt and enter a 0, a comma, a 1, and a closing parenthesis. Now press ENTER 5 times to generate a group of 0s and 1s. This simulates one trial (that is, one set of 5 coin flips). Carry out 4 trials and record your results in a table. Trial Number of heads???? Module 3 81 Lesson 1

2 Report your results to your teacher in order to combine everyone s results. Use the combined class data to complete the table. To find the relative frequency for an outcome, divide the frequency of the outcome by the total number of trials in the class and round to the nearest hundredth. Record the results in a table. Number of heads Frequency?????? Relative frequency?????? Enter the outcomes (0 through 5) into your calculator as list L 1. Enter the relative frequencies as list L. Make a histogram by turning on a statistics plot, selecting the histogram option, and using L 1 for Xlist and L for Freq as shown. Then set the viewing window as shown. Finally, press GRAPH to obtain a histogram like the one shown. Describe the shape of the probability distribution. Reflect 1. Discussion If you flipped a coin 5 times and got 5 heads, would this cause you to question whether the coin is fair? Why or why not? Explain 1 Displaying and Analyzing a Theoretical Probability Distribution Recall that a binomial experiment involves repeated trials where each trial has only two outcomes: success or failure. The probability of success on each trial is p, and the probability of failure on each trial is q = 1 - p. The binomial probability of r successes in n trials is given by P (X = r) = n C r p rq n-r. Example 1 Calculate all the theoretical probabilities for the given binomial experiment. Then draw a histogram of the probability distribution, observe its shape, and use it to find the specified probabilities. A binomial experiment consists of flipping a fair coin for 5 trials where getting heads is considered a success. Find the probability of getting 3 or more heads and the probability of getting at least 1 head. To calculate the probabilities, set n equal to 5 and let r range from 0 to 5 r in n C r p q n-r. Since the coin is fair, p = _ 1 and q = _ 1. Module 3 8 Lesson 1

3 Number of heads Theoretical probability 1 _ 5_ 10 _ 10 _ 5_ 1_ Create a histogram. The distribution is mounded and has symmetric tails, so it is a normal distribution. 10 The probability of getting 3 or more heads is: P (X 3) = P (X = 3) + P (X = 4) + P (X = 5) = _ 10 + _ 5 + _ 1 = _ 1 = 0.5 The probability of getting at least 1 head is: P (X 1) = 1 - P (X = 0) = 1 -_ 1 = _ A binomial experiment consists of flipping a biased coin for 5 trials where getting heads is considered a success. The coin lands heads up 75% of the time. Find the probability of getting 3 or more heads and the probability of getting at least 1 head. Probability Number of Heads To calculate the probabilities, set n equal to 5 and let r range from 0 to 5 in n C r p rq n-r. Since the coin is biased such that it lands heads up 75% of the time, p = 3 4 and q = 1_ 4. Number of heads Theoretical probability Create a histogram. Probability Number of Heads Module 3 83 Lesson 1

4 The distribution is mounded and has a tail to the left, so the distribution is skewed left. The probability of getting 3 or more heads is: P (X 3) = P (X = 3) + P (X = 4) + P (X = 5) = = 0.89 The probability of getting at least 1 head is: P (X 1) = 1 - P (X = 0) 1 = = Reflect. Why are the probabilities in the histogram you made in the Explore different from the probabilities given in the histogram from Part A? 3. For which coin, the fair coin in Part A or the biased coin in Part B, is flipping a coin 5 times and getting 5 heads more likely to occur? Explain. 4. Discussion Can you definitively conclude whether a coin that results in repeated heads when flipped is fair or biased? What might make you favor one conclusion over the other? Your Turn Calculate all the theoretical probabilities for the given binomial experiment. Then draw a histogram of the probability distribution, observe its shape, and use it to find the specified probabilities. 5. A binomial experiment consists of flipping a biased coin for 4 trials where getting heads is considered a success. The coin lands heads up 40% of the time. Find the probability of getting or more heads and the probability of getting fewer than 4 heads. Number of heads Theoretical probability????? Module 3 84 Lesson 1

5 Your Turn. A binomial experiment consists of flipping a biased coin for 4 trials where getting tails is considered a success. The coin lands heads up 40% of the time. Find the probability of getting or more tails and the probability of getting fewer than 4 tails. Number of heads Theoretical probability????? Elaborate 7. What is a random variable, and what makes a random variable discrete? 8. How can a histogram for a probability distribution be used to calculate a cumulative probability? 9. Essential Question Check-In What is a probability distribution for a discrete random variable? Evaluate: Homework and Practice 1. A spinner has three equal sections, labeled 1,, and 3. 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? 3 b. Make a table that shows all possible sums and the probability for each sum. c. Make a histogram of the probability distribution. d. What is the probability that the sum is not? How is this probability represented in the histogram? 1 Module 3 85 Lesson 1

6 . 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. Make a table that shows all possible absolute differences and the probability for each absolute difference. c. Is this probability distribution symmetric? Why or why not? d. Find the probability of getting a difference greater than A trick coin is designed to land heads up with a probability of 80%. You flip the coin 7 times. a. Make a table that shows all possible numbers of heads and the probability for each number of heads. b. Make a histogram of the probability distribution. c. What is the probability of getting or 7 heads? d. What is the probability of getting 4 or more heads? e. Which is greater, the probability of getting an even number of heads or the probability of getting an odd number of heads? f. 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? 4. You flip a coin 4 times in a row. The histogram shows the theoretical probability distribution for this situation. 1 Probability Number of Heads a. What is the probability of getting 3 or more heads? b. What is the probability of getting at most heads? c. How do you know that the coin is fair? Module 3 8 Lesson 1

7 5. A spinner has 4 equal sections that are labeled 1,, 3, and 4. You spin the spinner twice and find the sum of the numbers it lands on. Let X be a random variable that represents the sum of the numbers. a. Complete the table. Sum Frequency Probability??????? b. Make a histogram of the probability distribution. c. What is the probability of getting a sum of or more? d. Without actually calculating any probabilities, determine the relationship between P (X > 5) and P (X < 5). Explain your reasoning.. You roll number cubes at the same time. Let X be a random variable that represents the sum of the numbers rolled. a. Create a table to show the sums that are possible. In the table, make the row heads the numbers that are possible on one number cube, and the column heads the numbers that are possible on the other number cube. b. Create a table with row heads sum, frequency, and probability, as in Exercise 5. In the frequency row of the table, show the number of ways that you can get each sum. Then find the probability of each sum to complete the third row. c. Make a histogram of the probability distribution. d. What is the probability that you roll a sum of 5 or less? e. What is the probability that you roll a sum of 1 four times in a row? If this happened, would you question whether the number cubes are fair? Module 3 87 Lesson 1

8 7. A fair coin is flipped times. Match each specified probability on the left with its value on the right. (A value on the right may apply to more than one specified probability on the left.) A. The probability of getting at least 4 heads a.? B. The probability of getting no more than 1 head b.? C. The probability of getting 1 or heads c.? D. The probability of getting no more than heads d.? E. The probability of getting an even number of heads 7_ 4 1_ 4 _ 11 1_ F. The probability of getting an odd number of heads H.O.T. Focus on Higher Order Thinking 8. Represent Real-World Situations About 19.4% of the U.S. population that is 5 years old and over have a bachelor s degree only, and 10.5% have an advanced degree. a. Find the probability that of randomly selected people who are 5 years old or over, 4 have at least a bachelor s degree. b. Find the probability that of randomly selected people who are 5 years old or over, 4 do not have even a bachelor s degree. c. Suppose all of randomly selected people have advanced degrees. Would you question the probability model? Explain. 9. Justify Reasoning Describe a way to get fair results from a coin that you suspect is biased. Explain how you know that the process is fair. 10. Construct Arguments Use the formula P (X = r) = n C r p r q n-r for a binomial experiment to explain why the probability distribution for the number of heads obtained when a fair coin is flipped n times is symmetric. Lesson Performance Task According to the U.S. Census, in 010 the number of people 18 years old or over in the U.S. was 9.1 million, and of those people, 19.5 million were married. a. Find the probability that of 10 randomly selected people 18 years old or over, are married. b. Consider a survey where all 10 of the people surveyed are married. What conclusion might you draw about that survey? Module 3 88 Lesson 1