Table of Contents Introduction to Probability Experimental and Theoretical Sampling Word Problems Probability of Compound Events Measures of Center Mean Absolute Deviation Glossary Probability One way to express probability is to use a fraction. Probability of an event Number of favorable outcomes possible outcomes 1
Probability Probability Step 3: Put it all together to answer the question. The probability of flipping a nickel and landing on heads is: 1 2 Probability Probability Click to Reveal Click to Reveal Click to Reveal 2
Probability 1 probability of drawing an A from the bag? Click for hint 10 2 3 Click for hint Click for hint 3
4 5 6 7 what is the probability of picking a white t-shirt? 4
8 9 probability that the integer she chooses is a prime number? 10 Each of the hats shown below has colored marbles placed inside. Hat A contains five green marbles and four red marbles. Hat B contains six blue marbles and five red marbles. Hat C contains five green marbles and five blue marbles. If a student were to randomly pick one marble from each of these three hats, determine from which hat the student would most likely pick a green marble. Justify your answer. Determine the fewest number of marbles, if any, and the color of these marbles that could be added to each hat so that the probability of picking a green marble will be one-half in each of the three hats. Hat A contains five green marbles and four red marbles. Hat B contains six blue marbles and five red marbles. Hat C contains five green marbles and five blue marbles. A Hat B Hat C Hat 5
Outcomes Experimental Probability Experimental Probability Out of 31 golfers, you could expect 6 to make a hole-in-one on the last hole. Or there is a 19% chance of a golfer making a hole-in-one on the last hole. 6
Experimental Probability Experimental Probability Sally rolled a die 10 times and the results are shown Use this information to answer the following questions. # on Die Picture of Roll 0 fours 4 fives 1 six 11 What is the experimental probability of rolling a 5? 1/2 5/4 4/5 # on Die Picture of Roll 12 What is the experimental probability of rolling a 4? 1/2 # on Die Picture of Roll 5/4 0 2/5 0 fours 4/4 0 fours 4 fives 4 fives 1 six 1 six These are the results after 10 rolls of the die These are the results after 10 rolls of the die 7
13 Based on the experimental probability you found, if you rolled the die 100 times, how many sixes would you expect to get? 6 sizes 10 sixes # on Die Picture of Roll 14 Mike flipped a coin 15 times and it landed on tails 11 times. 12 sixes 60 sixes 0 fours 4 fives 1 six These are the results after 10 rolls of the die Theoretical Probability What is the theoretical probability of spinning green? Is this a fair probability? Theoretical Probability number of favorable outcomes total number of possible outcomes 8
Theoretical Probability Theoretical Probability 4 2 10 5 Theoretical Probability 15 What is the theoretical probability of picking a green marble? 1/8 7/8 1/7 1 9
16 What is the theoretical probability of picking a black marble? 1/8 7/8 1/7 0 17 What is the theoretical probability of picking a white marble? 1/8 7/8 1/4 1 18 What is the theoretical probability of not picking a white marble? 3/4 7/8 1/7 1 19 1/2 3 1/6 1 10
20 21 1/2 3 1/6 5/6 2/3 4 1/6 5/6 22 23 Seth tossed a fair coin five times and got five heads. 2/3 2 1/6 5/6 11
24 25 The spinner shown is divided into 8 equal sections. 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. 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 12
Answer the following: What is the theoretical probability for flipping a tail? A head? Compare the experimental probability to the theoretical probability for 10 experiments. Sampling Compare the experimental probability to the theoretical probability when the experiments for all of the students are considered? selected. If a sample is biased, then information obtained from it may not be reliable. Example mass transit, people at a train station are asked their opinion. Is this situation representative of the general population? 13
27 Food services at your school wants to increase the number 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. 28 The guidance counselors want to organize a career day. 29 situation produce a random sample? Explain your answer. sample? Explain your answer. 14
One way to estimate the number of wolves on a mountain is to use the capture-recapture method. Suppose this represents all the wolves on the mountain. Wildlife biologists first find some wolves and tag them. 15
Then they release them back onto the mountain. They wait until all the wolves have mixed together.then they find a second group of wolves and count how many are tagged. tagged wolves on mountain total wolves on mountain tagged wolves in second group total wolves in second group 16
30 31 32 realigned? 17
33 You survey 83 people leaving a voting site. 15 of them voted votes should Candidate A expect? 34 different types of shoes in Mr. Thomas' English class. Suppose that there are 300 students in the cafeteria. Predict how many would be wearing 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 three-month period. She determined that 28% of these people ordered for the menu with specials. Which statement about Josephine's sample is true? A The sample is the percentage of customers who order from the menu with specials. B The sample might not be representative of the popultation 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 18
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. 36 How many students participated in each survey? 37 chip should the student council purchase the most of? Regular BBQ Cheddar 19
38 student council should purchase. Word Problems 20
What are the actual values that will give you 19%? Remember sometimes it helps to turn a percent into a decimal prior to Click for hint solving the problem. Experimental Probability Experimental Probability Example 2 - Gardening 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. Solve this proportion using cross products. 21
Experimental Probability 39 40 Tom was at bat 50 times and hit the ball 10 times. Estimate 41 jack from a deck of cards? 22
42 43 Mark rolled a 3 on a die for 7 out of 20 rolls. What is the theoretical probability for rolling a 3? 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 45 2/52 4/52 2/13 8/52 23
46 4/13 13/52 2/13 16/52 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. Any number from 1 through 6 represents a person who buys a movie ticket and popcorn. Any number from 7 through 9 or 0 represents a person who buys only a movie ticket. Use info for next two questions. 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 randomly-chosen people who buy movie tickets also buys 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. B Exactly one person in a group of three randomly-chosen people who buy movie tickets also buys popcorn. C Exactly two people in a group of three randomly-chosen people who buy movie tickets also buy popcorn. D All three people in a group of three randomly-chosen people who buy movie tickets also buy popcorn. 24
Probability of Compound Events For the probability of compound events, first - decide if the two events are independent or dependent. Probability of Compound Events When the outcome of one event does not affect the outcome of another event, the two events are independent. Use formula: Click to go to 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? When the outcome of one event affects the outcome of another event, the two events are dependent. Use formula: Dependent Events 25
Dependent Example Select a card from a deck of cards, do not replace it in the deck, shuffle the deck, and select a second card. Independent & Dependent Examples Try to name some other independent and dependent events. What is the probability that you will pick a 6 and then a king? Notice your demominator when down by 1. Why? 49 The names of 6 boys and 10 girls from your class are put in a hat. What is the probability that the first two names chosen will both be boys? (w/o replacement) 50 A lottery machine generates numbers randomly. Two numbers between 1 and 9 are generated. What is the probability that both numbers are 5? 26
51 The TV repair person is in a room with 20 broken TVs. Two sets have broken wires and 5 sets have a faulty computer chip. What is the probability that the first TV repaired has both problems? 52 What is the probability that the first two cards drawn from a full deck are both hearts? (without replacement) 53 A spinner containing 5 colors: red, blue, yellow, white and green is spun and a die, numbered 1 through 6, is rolled. What is the probability of spinning green and rolling a two? 54 A drawer contains 5 brown socks, 6 black socks, and 9 navy blue socks. The power is out. What is the probability that Sam chooses two socks that are both black? 27
55 56 R B G What is the probability that it will land in section G the first time and then in section B the second time? Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, 57 Measures of Center 2 Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, 28
Generalizations 58 Alexis chose a random sample of 10 jars of almonds from each of size. She counted the number of almonds in each jar. Her results are shown in the plots. Sometimes we can make general statements about a set of data as shown in this first question. A The number of almonds in jars from those from Brand Y. B The number of almonds in jars from Brand X tends to be greater and less consistent than those from Brand Y. C The number of almonds in jars from Brand X tends to be fewer and more consistent than those from Brand Y. D The number of almonds in jars from Brand X tends to be fewer and less consistent than those from Brand Y. From PARCC EOY sample test calculator #7 Generalizations Measures of Center - Vocabulary Review Other times we will make statements about the data based on measure of center and variation that we can calculate. This will be the topics for the rest of this chapter. 29
Measures of Center Joey wanted to convince his mom to give him some money for a snack from the concession stand. Below are the prices of the different snacks. $1.75, $0.75, $1.25, $0.75, $2.50, $2.00 Mean Example What is the mean of this data set? $1.75, $0.75, $1.25, $0.75, $2.50, $2.00 Step 1: Add up all of the numbers. 1.75 + 0.75 + 1.25 + 0.75 + 2.50 + 2.00 = 9.00 Step 2: Divide the sum by the number of items listed. 9.00 / 6 = 1.50 The mean cost of concession stand snacks is $1.50. Median Example What is the median of this data set? $1.75, $0.75, $1.25, $0.75, $2.50, $2.00 Step 1: Order the numbers from least to greatest. 0.75, 0.75, 1.25, 1.75, 2.00, 2.50 Step 2: Find the middle value. 0.75, 0.75, 1.25, 1.75, 2.00, 2.50 The median cost of concession stand snacks is $1.50. What do you do Find the mini mean when you have two numbers left? (click) Mode Example What is the mode of this data set? $1.75, $0.75, $1.25, $0.75, $2.50, $2.00 Step 1: Look for the number that appears most often. 1.75, 0.75, 1.25, 0.75, 2.50, 2.00 The mode cost of concession stand snacks is $0.75. 30
Measures of Center How can Joey use this information to ask his mom for money? $1.75, $0.75, $1.25, $0.75, $2.50, $2.00 Measures of Center Use the dot plots to compare the 2 samples. $1.50 $1.50 $0.75 Measures of Center Measures of Center Find the mean, median, and mode for the sample of girls. Find the mean, median, and mode for the sample of boys. 31
Measures of Center 59 What is the mean of the stem-and-leaf plot? Now compare the two measures of center. Girls Boys Mean 88.5 40.5 Median 90 30 Mode 60 30 and 60 Make a statement about the average time spent texting daily by 7th grade students. 60 What is the median of the stem-and-leaf plot? 61 What is the mode of the stem-and-leaf plot? 32
62 What is the mean of the stem-and-leaf plot? 63 What is the median of the stem-and-leaf plot? 64 What is the mode of the stem-and-leaf plot? Measures of Center Use the dot plots to find the measures of center. 1st Period Scores 33
Measures of Center Use the dot plots to find the measures of center. Measures of Center Write a statement comparing the averages of Miss M's 1st period class scores to her 8th period class scores. 8th Period Scores Measures of Variation - Vocabulary Review Range - The difference between the greatest data value and the least data value. Quartiles - are the values that divide the data in four equal parts. Lower (1st) Quartile (Q1) - The median of the lower half of the data. Upper (3rd) Quartile (Q3) - The median of the upper half of the data. Interquartile Range - The difference of the upper quartile and the lower quartile. (Q3 - Q1) 34
Interquartile Range Interquartile Range 5 median Interquartile Range Interquartile Range Upper Lower - Quartile Quartile = Interquartile Range Quartile median 35
Sample Range Greatest - Least = Range 1 Least 2 Lower Quartile 5 Median 8 Upper Quartile 9 Greatest click just above the number line to reveal 65 What is the median of the data set? 66 What is the interquartile range using the given information? Lower Quartile = 6 Median = 7 Upper Quartile = 10 36
67 What is the range for the following data set? 3, 5, 10, 4, 2, 2, 1 68 What is the interquartile range for the following data set? 3, 5, 10, 4, 2, 2, 1 Mean Absolute Deviation - Vocabulary Mean Absolute Deviation 37
Mean Absolute Deviation Find the mean absolute deviation of the following data. Step 1: Find the mean. Quiz Scores 65, 75, 90, 90, 100 65 + 75 + 90 + 90 + 100 = 420 = 84 5 5 Score Mean Absolute Deviation Step 2: Find the absolute deviation. To do this you need to subtract the mean and each data point. Then take the absolute value of each difference. Deviation from mean Absolute deviation from mean 65 65-84 = -19-19 = 19 75 75-84 = -9-9 = 9 90 90-84 = 6 6 = 6 90 90-84 = 6 6 = 6 100 100-84 = 16 16 = 16 Mean Absolute Deviation Step 3: Find the mean absolute deviation (MAD). To do this find the mean using the absolute deviation numbers. Comparing Two Data Sets The number of goals scored by the players on the boys' and girls' LAX teams are displayed below. Absolute deviation from mean -19 = 19-9 = 9 6 = 6 6 = 6 16 = 16 19 + 9 + 6 + 6 + 16 5 = 56 = 11.2 5 The MAD is 11.2 points. Compare the variability of the mean goals scored for both teams. 38
Comparing Two Data Sets Step 1: Find the mean for each team. Comparing Two Data Sets Step 2: Find the absolute deviations. Goals Mean Deviation Absolute Mean Dev. Goals Mean Deviation Absolute Mean Dev. Comparing Two Data Sets Comparing Two Data Sets Step 3: Find the mean absolute deviations. Comparison Statements 1.25 = 1.25 The variability is equal for both the boys and girls LAX teams. On average, the boy players scored 1 more goal than the girl players. (How do you know this?) 39
Comparing Two Data Sets 69 Use the following data to answer the next seven questions. the Hunger Games? Pages per Chapter in Hunger Games 70 What is the mean number of pages per chapter in 71 What is the difference of the means? 40
72 for Hunger Games? 73 for Twilight? 74 75 On average, there are pages per chapter in the Hunger Games than in Twilight. Hunger Games A more B less 41
Glossary When every possible sample of the same size does not have an equal chance of being selected. Capture-Recapture Method A method of sampling that is used to try and estimate the entire population. A sample of animals are caught, tagged, and then released into the wild. Later a second sample of animals are caught to compute using a ratio the amount of tagged animals to the population as a whole. A combination of two or more simple events. 42
When the outcome of one affects the outcome of another event. When all the outcomes have the same chance of occurring. Spinner The ratio of the number of times an event occurs to the total number of times that the activity is performed. An experiment with equally likely outcomes. 43
When the outcome of one event does not affect the outcome of another event. The difference of the upper quartile and the lower quartile. - - Lower (1st) Quartile The median of the lower half of data. Average The sum of the data values divided by the number of items. The mean is 3. 44
Deviation The average distance between each data value and the mean. The middle data value when the values are written in numerical order. 2,2,3,4,4 =4 5= The data value that occurs the most often., 6,, 4, 6, 3, 8 There is no mode. All m&ms in a bag All types of dogs in a dog park wearing glasses in a classroom 45
Probability The ratio of the number of favorable outcomes to the total number of possible outcomes. The values that divide the data in four equal parts. 1 2 Many Forms! Random Sample Unbiased - Every possible sample of the same size has an equal chance of being selected The difference between the greatest data value and the least data value., 4, 7, 1, 5, 9,, 9, 10, The range The range The range 46
A part of a group. The ratio of the number of outcomes in an event to the total number of possible outcomes. random or unbiased only red m&ms in a bag only poodles in a dog only girls wearing glasses in a classroom number of favorable outcomes total number of possible outcomes Find the probability of getting tails when you flip a coin. The median of the upper half of data. 47