17 What is the theoretical probability of picking a white marble? A W R Y B G Answer Y B C C W R D
18 What is the theoretical probability of not picking a white marble? W A R Y G B Y B Answer W C R A D
19 What is the theoretical probability of rolling a three? A B Answer C D
20 What is the theoretical probability of rolling an odd number? A B Answer A C D
21 What is the theoretical probability of rolling a number less than 5? A B Answer C A D
22 What is the theoretical probability of not rolling a 2? A B C Answer D D
23 Seth tossed a fair coin five times and got five heads. The probability that the next toss will be a tail is A B C Answer D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
24 Which inequality represents the probability, x, of any event happening? A B C Answer D D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011
25 The spinner shown is divided into 8 equal sections. 1 2 1 2 4 3 2 5 Answer The arrow on this spinner is spun once. What is the probability that the arrow will land on a section labeled with a number greater than 3? Enter only your fraction. From PARCC EOY sample test calculator #1
26 Reagan will use a random number generator 1,200 times. Each result will be a digit form 1 to 6. Which statement best predicts how many times the digit 5 will appear among the 1,200 results? A It will appear exactly 200 times. Answer B It will appear close to 200 times but probably not exactly 200 times. C It will appear exactly 240 times. D It will appear close to 240 times but probably not exactly 240 times. From PARCC EOY sample test calculator #17
Class Activity • Each student flips a coin 10 times and records the number of heads and the number of tail outcomes. • Each student calculates the experimental probability of flipping a tail and flipping a head. • Use the experimental probabilities determined by each student to calculate the entire class's experimental probability for flipping a head and flipping a tail.
Class Activity What is the theoretical probability for flipping a tail? A head? Compare the experimental probability to the theoretical probability for 10 experiments. Compare the experimental probability to the theoretical probability when the experiments for all of the students are considered?
Sampling Return to Table of Contents
Sampling Your task is to count the number of whales in the ocean or the number of squirrels in a park. How could you do this? What problems might you face? A sample is used to make a prediction about an event or gain information about a population. A whole group is called a POPULATION. A part of a group is called a SAMPLE.
Sampling A sample is considered random (or unbiased) when every possible sample of the same size has an equal chance of being selected. If a sample is biased , then information obtained from it may not be reliable. Answer Example: To find out how many people in New York feel about mass transit, people at a train station are asked their opinion. Is this situation representative of the general population?
Sampling Determine whether the situation would produce a random sample. You want to find out about music preferences of people living in your area. You and your friends survey every tenth person who enters the mall nearest you. Answer
27 Food services at your school wants to increase the number of students who eat hot lunch in the cafeteria. They conduct a survey by asking the first 20 students that enter the cafeteria lunch line to determine the students' preferences for hot lunch. Is this survey reliable? Explain your answer. Yes Answer No
28 The guidance counselors want to organize a career day. They will survey all students whose ID numbers end in a 7 about their grades and career counseling needs. Would this situation produce a random sample? Explain your answer. Yes Answer No Yes
29 The local newspaper wants to run an article about reading habits in your town. They conduct a survey by asking people in the town library about the number of magazines to which they subscribe. Would this produce a random sample? Explain your answer. Yes Answer No No
Sampling How would you estimate the size of a crowd? What methods would you use? Could you use the same methods to estimate the number of wolves on a mountain?
Sampling A whole group is called a population. A part of a group is called a SAMPLE. When biologists study a group of wolves, they are choosing a sample. The population is all the wolves on the mountain. Population Sample
Sampling One way to estimate the number of wolves on a mountain is to use the capturerecapture method . Suppose this represents all the wolves on the mountain.
CaptureRecapture Method Wildlife biologists first find some wolves and tag them.
CaptureRecapture Method Then they release them back onto the mountain.
CaptureRecapture Method They wait until all the wolves have mixed together.Then they find a second group of wolves and count how many are tagged.
CaptureRecapture Method Biologists use a proportion to estimate the total number of wolves on the mountain: tagged wolves on mountain tagged wolves in second group = total wolves on mountain total wolves in second group For accuracy, they will often conduct more than one recapture. There are 36 wolves on the mountain
CaptureRecapture Method Try This: Biologists are trying to determine how many fish are in the Rancocas Creek. They capture 27 fish, tag them and release them back into the Creek. 3 weeks later, they catch 45 fish. 7 of them are tagged. How many fish are in the creek? Click There are 174 fish in the river.
CaptureRecapture Method Try This: 315 out of 600 people surveyed voted for Candidate A. How many votes can Candidate A expect in a town with a population of 1500? Answer
30 Eight hundred sixty out of 4,000 people surveyed watched Dancing with the Stars. How many people in the US watched if there are 93.1 million people? Answer
31 Six out of 150 tires need to be realigned. How many out of 12,000 are going to need to be realigned? Answer
32 You are an inspector. You find 3 faulty bulbs out of 50. Estimate the number of faulty bulbs in a lot of 2,000. Answer 120 faulty bulbs
33 You survey 83 people leaving a voting site. 15 of them voted for Candidate A. If 3,000 people live in town, how many votes should Candidate A expect? Answer about 542 votes
34 The chart shows the number of people wearing different types of shoes in Mr. Thomas' English class. Suppose that there are 300 students in the cafeteria. Predict how many would be wearing hightop sneakers. Explain your reasoning. Number of Shoes Students Lowtop 12 Answer sneakers 75 students Hightop 7 sneakers Sandals 3 Boots 6
35 Josephine owns a diner that is open every day for breakfast, lunch, and dinner. She offers a regular menu and a menu with specials for each of the three meals. She wanted to estimate the percentage of her customers that order form the menu with specials. She selected a random sample of 50 customers who had lunch at her diner during a threemonth period. She determined that 28% of these people ordered for the menu with specials. Which statement about Josephine's sample is true? Answer A The sample is the percentage of customers who order from the menu with specials. B The sample might not be representative of the population because it only included lunch customers. C The sample shows that exactly 28% of Josephine's customers order from the menu with specials. D No generalizations can be made from this sample, because the sample size of 50 is too small. From PARCC EOY sample test calculator #13
Multiple Samples The student council wanted to determine which lunch was the most popular among their students. They conducted surveys on two random samples of 100 students. Make at least two inferences based on the results. Student Sample Hamburgers Tacos Pizza Total 14 74 #1 12 100 11 77 100 #2 12 Click • Most students prefer pizza. • More people prefer pizza than hamburgers and tacos combined.
Multiple Samples The NJ DOT (Department of Transportation) used two random samples to collect information about NJ drivers. The table below SUV's are the most popular shows what type of vehicles were being driven. Make at least two vehicles among NJ drivers. inferences based on the results of the data. Cars are the second most Answer popular vehicles among NJ drivers. Mini Motorcycles Total Cars SUVs Driver Sample Vans SUV's are at least twice as 100 37 43 12 #1 8 popular as mini vans and motorcycles combined. 46 100 33 11 10 #2
The student council would like to sell potato chips at the next basketball game to raise money. They surveyed some students to figure out how many packages of each type of potato chip they would need to buy. For home games, the expected attendance is approximately 250 spectators. Use the chart to answer the next three questions. Student Regular BBQ Cheddar Sample #1 8 10 7 #2 8 11 6
36 How many students participated in each survey? Student Regular BBQ Cheddar Sample #1 8 10 7 #2 8 11 6 Answer 25
37 According to the two random samples, which flavor potato chip should the student council purchase the most of? Regular A BBQ B Answer C Cheddar B Student Regular BBQ Cheddar Sample #1 8 10 7 #2 8 11 6
38 Use the first random sample to evaluate the number of packages of cheddar potato chips the student council should purchase. Student Regular BBQ Cheddar Sample #1 8 10 7 Answer #2 8 11 6 70
Word Problems Return to Table of Contents
Word Problems Erica loves soccer! The ladies' coach tells Erica that she scored 19% of her attempts on goal last season. This season, the coach predicts the same percentage for Erica. Erica reports she attempted approximately 1,100 shots on goal last season. Her coach suggests they estimate the number of goals using experimental probability. Click What do you know about percentages 19 shots made 100 shots attempted = 19% to figure out the relationship of goals scored to goals attempted? number of times the outcome happened Experimental Probability = number of times experiment was repeated number of goals Click Erica's Experimental = Probability number of attempts Click
Word Problem Let's estimate the number of goals Erica scored. Erica takes 1,100 shots on Erica makes 19% of her shots goal. on goal. About what percent would be a About how many attempts good estimate to use? did Erica take? click 1,100 is very close to click is very close to 1,000. So we will estimate so she makes about 20% of that Erica has about 1,000 attempts. her shots on goal.
Word Problem Erica wants to find 20% of 1,000. Her math looks like this: Click Erica figures she made about 200 of her shots on goal.
Word Problem What are the actual values Challenge that will give you 19%? Answer Click Remember sometimes it helps to turn a percent into a decimal prior to solving the problem.
Experimental Probability Last year, Lexi planted 12 tulip bulbs, but only 10 of them bloomed. This year she intends to plant 60 tulip bulbs. Use experimental probability to predict how many bulbs will bloom. Solve this proportion by equivalent fractions. Based on her experience last year, Lexi can expect 50 out of 60 tulips to bloom.
Experimental Probability Today you attempted 50 free throws and made 32 of them. Use experimental probability to predict how many free throws you will make tomorrow if you attempt 75 free throws. Solve this proportion using cross products. Based on your performance yesterday,you can expect to make 48 free throws out of 75 attempts.
Experimental Probability Now, its your turn. Calculate the experimental probability for the number of goals. Experimental Number of Number of goals Probability attempts 100 30 1000 600 500 150 1600 2000
39 Tom was at bat 50 times and hit the ball 10 times. What is the experimental probability for hitting the ball? Answer
40 Tom was at bat 50 times and hit the ball 10 times. Estimate the number of balls Tom hit if he was at bat 250 times. Answer 50
41 What is the theoretical probability of randomly selecting a jack from a deck of cards? Answer
42 Mark rolled a 3 on a die for 7 out of 20 rolls. What is the experimental probability for rolling a 3? Answer
43 Mark rolled a 3 on a die for 7 out of 20 rolls. What is the theoretical probability for rolling a 3? Answer
44 Some books are laid on a desk. Two are English, three are mathematics, one is French, and four are social studies. Theresa selects an English book and Isabelle then selects a social studies book. Both girls take their selections to the library to read. If Truman then selects a book at random, what is the probability that he selects an English book? Answer
45 What is the probability of drawing a king or an ace from a standard deck of cards? A B Answer C C D
46 What is the probability of drawing a five or a diamond from a standard deck of cards? A B Answer A C D
Movie Theater Lindsey would like to know the number of people at a movie theater that will buy a movie ticket and popcorn. Based on past data, the probability that a person who is selected at random from those that buy movie tickets and also buy popcorn is 0.6. Lindsey designs a simulation to estimate the probability that exactly two in a group of three people selected randomly at a movie theater will buy both a movie ticket and popcorn. For the simulation Lindsey used a number generator that generates random numbers. 266 342 847 672 567 • Any number from 1 through 6 represents a person who buys a 268 252 465 429 573 movie ticket and popcorn. 100 818 139 730 910 • Any number from 7 through 9 or 0 494 922 155 585 426 represents a person who buys only a movie ticket. 593 903 556 981 966 • Use info for next two questions. 491 186 865 044 147 From PARCC EOY sample test calculator #3
47 Part A In the simulation, one result was "100". What does this result simulate? A No one in a group of three randomlychosen people who buy movie tickets also buys popcorn. B Exactly one person in a group of three randomlychosen Answer people who buy movie tickets also buys popcorn. C Exactly two people in a group of three randomlychosen people who buy movie tickets also buy popcorn. D All three people in a group of three randomlychosen people who buy movie tickets also buy popcorn.
48 Part B Use the results of the simulation to estimate the probability that exactly two of three people selected at random from those who buy movie tickets will also buy popcorn. Answer
Probability of Compound Events Return to Table of Contents
Probability of Compound Events For the probability of compound events , first decide if the two events are independent or dependent. When the outcome of one event does not affect the outcome of another event, the two events are independent . Use formula: Probability (A and B) = Probability (A) Probability (B)
Independent Example Select a card from a deck of cards, replace it in the deck, shuffle the deck, and select a second card. What is the probability that you will pick a 6 and then a king? P (6 and a king) = P(6) P(king)
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