10-3 Probability Distributions

Transcription

1 Identify the random variable in each distribution, and classify it as discrete or continuous. Explain your reasoning. 1. the number of pages linked to a Web page The random variable X is the number of pages linked to a Web page. The pages are countable, so X is discrete. 3. the amount of precipitation in a city per month The random variable X is the amount of precipitation in a city per month. Precipitation can be anywhere within a certain range. Therefore, X is continuous. 5. X represents the sum of the values of two spins of the wheel. a. Construct a relative-frequency table showing the theoretical probabilities. b. Graph the theoretical probability distribution. c. Construct a relative-frequency table for 100 trials. d. Graph the experimental probability distribution. e. Find the expected value for the sum of two spins of the wheel. f. Find the standard deviation for the sum of two spins of the wheel. a. There are 64 total outcomes possible for two spins of the wheel. List each possible sum in the first column. List the number of times the sum can occur in the second column. For example, a sum of 4 can only occur when a 2 is spun both times and a sum of 6 can occur when a 2 and 4 are spun or when a 4 and 2 are spun. To determine the relative frequency, or theoretical probability, of each outcome, divide the frequency by 64. b. The graph shows the probability distribution for the sum of two spins X. The bars are separated on the graph because the distribution is discrete (no other values of X are possible). c. Create a spinner equivalent to the one in the book or use a random number generator to complete the simulation and create a simulation tally sheet. Then determine the frequency and relative frequency. esolutions Manual - Powered by Cognero Page 1

2 d. The expected value is e. Using the theoretical probability distribution, multiply each sum by the corresponding relative frequency. Then find the sum of those values. Note the last column is the sums converted to a common denominator. f. Using the table from part e and E(X) = 13.5, subtract the expected value from each outcome. Square each difference. Then multiply by the each corresponding probability. Finally, find the sum of these values. The values in the last column are rounded to the nearest ten-thousandth. esolutions Manual - Powered by Cognero Page 2

3 11. SNOW DAYS The following probability distribution lists the probable number of snow days per school year at North High School. Use this information to determine the expected number of snow days per year. Find the sum of the weighted values of each variable. E(X) = Σ[X P(X)] = 3.34 The standard deviation is about Identify the random variable in each distribution, and classify it as discrete or continuous. Explain your reasoning. 7. the number of diggs (or likes ) for a Web page The random variable X is the number of diggs for a web page. The diggs are countable, so X is discrete. 9. the number of files infected by a computer virus The random variable X is the number of files infected by a computer virus. The files are countable, so X is discrete. 13. RAFFLES The table shows the probability distributi for a raffle if 100 tickets are sold for $1 each. There 1 prize for $20, 5 prizes for $10, and 10 prizes for $5. a. Graph the theoretical probability distribution. b. Find the expected value. c. Interpret the results you found in part b. What can you conclude about the raffle? a. Use the given probabilities to create the distributio The graph shows the probability distribution for the prize amounts X. The bars are separated on the grap because the distribution is discrete (no other values o X are possible). esolutions Manual - Powered by Cognero Page 3

4 a common denominator. b. List each prize value X along with the correspondi relative frequency P(X). Find X P(X). Then find the sum of those values.. X P(X) XP(X) Total 1.20 The expected value of winnings is $1.20. c. Sample answer: The expected value is positive, so person buying a ticket can expect to win $0.20 even after the cost of the ticket is considered. Thus, a pers would want to participate in this raffle. On the other hand, this raffle is guaranteed to lose money for the organizers and they should change the distribution of prizes or not do the raffle. 15. BASKETBALL The distribution below lists the probability of the number of major upsets in the first round of a basketball tournament each year. The expected number of upsets is or about b. Using the table from part a and E(X) =, subtract the expected value from each outcome. Square each difference. Then multiply by each corresponding probability. Finally, find the sum of these values. The values in the third and column are rounded to the nearest ten-thousandth. You can avoid calculation errors by using a graphing calculator. Enter the first column as L1 and the second column as L2. Set L3 equal to [L1 (139/32)] 2. Set L4 equal to L3 L2. Then find the sum of L4. a. Determine the expected number of upsets. Interpret your results. b. Find the standard deviation. c. Construct a relative-frequency table for 50 trials. d. Graph the experimental probability distribution. a. Multiply the value of X by the corresponding relative frequency P(X). Then find the sum of those values. Note the last column is the sums converted to esolutions Manual - Powered by Cognero Page 4

5 0 c. Use a random number generator to complete the simulation and create a simulation tally sheet. One challenge is identifying what numbers to use. First, convert P(X) to a common denominator for each value of X. Then we can let the numerators dictate the values that represent each number of upsets. 17. DECISION MAKING Carmen is thinking about investing $10,000 in two different investment funds. The expected rates of return and the corresponding probabilities for each fund are listed below. Compare the two investments using the expected value and standard deviation. Which investment would you advise Carmen to choose, and why? On your graphing calculator, select randint(1, 32, 50) for 50 trials. Then determine the frequency and relative frequency. Find the expected value of each investment. Multiply the possible value of each fund by the associated probability. A profit is a positive value while a loss is a negative value. Fund A: d. Use the given probabilities to create the distribution. The graph shows the probability distribution for the number of upsets X. The bars are separated on the graph because the distribution is discrete (no other values of X are possible). Fund B: esolutions Manual - Powered by Cognero Page 5

6 19. CCSS CRITIQUE Liana and Shannon each created a probability distribution for the sum of two spins on the spinner. Is either of them correct? Explain your reasoning. The expected values of Funds A and B are $595 and $540, respectively. Calculate each standard deviation. Fund A: Fund B: The expected value of Funds A and B is $595 and $540, respectively. The standard deviation for Fund A is about 951.6, while the standard deviation for Fund B is about Since the standard deviations are about the same, the funds have about the same amount of risk. Therefore, with a higher expected value, Fund A is the better investment. Sample answer: Liana; Shannon didn t consider every scenario in determining the total probability. For example, in calculating the probability of a sum of 5, she considered spinning a 3 then a 2, but not a 2 then a OPEN ENDED Create a discrete probability distribution that shows five different outcomes and their associated probabilities. Ensure that each outcome is independent of the others and that the sum of the probabilities of the outcomes is one. Sample answer: A spinner with 5 equal-sided areas shaded red, blue, yellow, green, and brown. esolutions Manual - Powered by Cognero Page 6

7 23. OPEN ENDED Provide examples of a discrete probability distribution and a continuous probability distribution. Describe the differences between them. Sample answer: A discrete probability distribution can be the uniform distribution of the roll of a die. In this type of distribution, there are only a finite number of possibilities. A continuous probability distribution can be the distribution of the lives of 400 batteries. In this distribution, there are an infinite number of possibilities. 25. GRIDDED RESPONSE The height f (x) of a bouncing ball after x bounces is represented by f (x) = 140(0.8) x. How many times higher is the first bounce than the fifth bounce? Substitute 1 and 5 for x and evaluate. 27. GEOMETRY Find the area of the shaded portion of the figure to the nearest square inch. F 79 G 94 H 589 J 707 The central angle of the shaded region is 300º. The area of a sector is. The area of the shaded region is in 2. Option H is the correct answer. Therefore, the first bounce is about 2.4 times higher than the fifth bounce. 29. ARTICLES Peter and Paul each write articles for an online magazine. Their employer keeps track of the number of likes received by each article. a. Use a graphing calculator to create a histogram for each data set. Then describe the shape of each distribution. b. Compare the distributions using either the means and standard deviations or the five-number summaries. Justify your choice. a. Peter s Articles: First, press STAT ENTER and enter each data value. Then, press 2ND [STAT PLOT] ENTER ENTER and choose. Finally, adjust the window to esolutions Manual - Powered by Cognero Page 7

8 the dimensions shown. Paul s Articles: The majority of the data are on the left of the mean, so the distribution is positively skewed. Paul s Articles: First, press STAT ENTER and enter each data value in L2. Then, press 2ND [STAT PLOT] ENTER ENTER and choose. Finally, adjust the window to the dimensions shown. The data are evenly distributed about the median, so the distribution is symmetric. b. One distribution is skewed, so use the five-number summaries. For the first distribution, press STAT ENTER ENTER and scroll down to view the five-number summary. For the second distribution, press STAT ENTER ENTER L2 and scroll down to view the five-number summary. The range for Peter s articles is 64, and the range for Paul s articles is 53. However, the upper quartile for Peter s is 33, while the lower quartile for Paul s is 34. This means that 75% of Paul s articles have more likes (and are more popular) than 75% of Peter s articles. Therefore, we can conclude that Paul s articles are more popular overall. Determine whether the situation calls for a survey, an observational study, or an experiment. Explain your reasoning. 31. You want to find voters opinions on recent legislation. This is a survey because data are collected from responses to the question. Find the first five terms of each geometric sequence described. 33. = 0.5, r = 2.5 Peter s Articles: The first five terms are 0.5, 1.25, 3.125, and esolutions Manual - Powered by Cognero Page 8

9 35. = 12, 37. = 80, The first five terms are 80, 100, 125, The first five terms are 12, 4,.. Solve each equation. Check your solutions. 39. esolutions Manual - Powered by Cognero Page 9

10 41. Expand each power. 43. (m + n) 4 esolutions Manual - Powered by Cognero Page 10