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This material is made freely available at and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: New Jersey Center for Teaching and Learning Progressive Mathematics Initiative

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th Grade Math Statistics & Probability

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Setting the PowerPoint View Use Normal View for the Interactive Elements To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: On the View menu, select Normal. Close the Slides tab on the left. In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen. On the View menu, confirm that Ruler is deselected. On the View tab, click Fit to Window. On the View tab, click Slide Master | Page Setup. Select On-screen Show (4:3) under Slide sized for and click Close Master View. On the Slide Show menu, confirm that Resolution is set to 1024x768. Use Slide Show View to Administer Assessment Items To administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 9 for an example.)

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STATISTICS & PROBABILITY Introduction to Probability Fundamental Counting Principle Experimental and Theoretical Probabilities of Mutually Exclusive and Overlapping Events Word Problems Permutations and Combinations Probability of Compound Events Complementary Events Click on a topic to go to that section. Common Core: 7.SP.1-8 Sampling Comparing Two Populations

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Sampling Return to table of contents

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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.

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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. 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? No. The sample only includes people who take the train and does not include people who may walk, drive, or bike.

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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.

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1Food 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 to determine the students' preferences for hot lunch. Is this survey reliable? Explain your answer. Yes No A B

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2The 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 No A B

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3The 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 No A B

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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?

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One way to estimate the number of wolves on a mountain is to use the CAPTURE - RECAPTURE METHOD.

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Suppose this represents all the wolves on the mountain.

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Wildlife biologists first find some wolves and tag them.

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Then they release them back onto the mountain.

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They wait until all the wolves have mixed together. Then they find a second group of wolves and count how many are tagged.

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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. = 8 2 w 9 2w = 72 w = 36 = There are 36 wolves on the mountain

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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? 27 7 f 45 27(45) = 7f 1215 = 7f = f = There are 174 fish in the river

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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

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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?

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4860 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?

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5Six out of 150 tires need to be realigned. How many out of 12,000 are going to need to be realigned?

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6You are an inspector. You find 3 faulty bulbs out of 50. Estimate the number of faulty bulbs in a lot of 2,000.

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7You 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?

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8The 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 high-top sneakers. Explain your reasoning. Low-top sneakers12 High-top sneakers7 Sandals3 Boots6 Shoes Number of Students

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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 SampleHamburgersTacosPizzaTotal # # Most students prefer pizza. More people prefer pizza than hamburgers and tacos combined.

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Try This! The NJ DOT (Department of Transportation) used two random samples to collect information about NJ drivers. The table below shows what type of vehicles were being driven. Make at least two inferences based on the results of the data. Driver SampleCarsSUVsMini VansMotorcyclesTotal # #

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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 following questions. Student Sample RegularBBQCheddar #18107 #28116

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9How many students participated in each survey?

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10According to the two random samples, which flavor potato chip should the student council purchase the most of? ARegular BBBQ CCheddar

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11Use the first random sample to evaluate the number of packages of cheddar potato chips the student council should purchase.

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Comparing Two Populations Return to table of contents

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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) Mean absolute deviation - the average distance between each data value and the mean.

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Example: Victor wants to compare the mean height of the players on his favorite basketball and soccer teams. He thinks the mean height of the players on the basketball team will be greater but does not know how much greater. He also wonders if the variability of heights of the athletes is related to the sport they play. He thinks that there will be a greater variability in the heights of soccer players as compare to basketball players. He uses the rosters and player statistics from the team websites to generate the following lists. Height of Soccer Players (inches) 73, 73, 73, 72, 69, 76, 72, 73, 74, 70, 65, 71, 74, 76, 70, 72, 71, 74, 71, 74, 73, 67, 70, 72, 69, 78, 73, 76, 69 Height of Basketball Players (inches) 75, 73, 76, 78, 79, 78, 79, 81, 80, 82, 81, 84, 82, 84, 80, 84 from

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Victor creates two dot plots on the same scale x x xxxxxx xxxxxxxx xxxxxxxxxxxx xxxxxx xxxxxx xxxxxxxx xxxxxx x Height of Soccer Players (inches) x xx xxxx xxxx xxxx xxxx xxxx xxxxxx Height of Basketball Players (inches)

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x x xxxxxx xxxxxxxx xxxxxxxxxxxx xxxxxx xxxxxx xxxxxxxx xxxxxx x Height of Soccer Players (inches) x xx xxxx xxxx xxxx xxxx xxxx xxxxxx Height of Basketball Players (inches) Victor notices that although generally the basketball players are taller, there is an overlap between the two data sets. Both teams have players that are between 73 and 78 inches tall.

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To express the difference between centers of two data sets as a multiple of a measure of variability, first find the difference between the centers. *Recall: The difference between the means is – = Divide the difference by the mean absolute deviations of each data set ÷ 2.14 = = 3.07 The difference of the means (7.68) is approximately 3 times the mean absolute deviations.

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Use the following data to answer the next set of questions. Pages per Chapter in Hunger Games x xxxxxxxx xxxxxx xxxxxxxxxxxxxx xxxxxx xxxxxxxx x xxxxxx x xxxxxx x x xxxx xxxx xxxx xxxx x xxxx x xxxxxx xxxx Pages per Chapter in Twilight

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12What is the mean number of pages per chapter in the Hunger Games?

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13What is the mean number of pages per chapter in Twilight?

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14What is the difference of the means?

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15What is the mean absolute deviation of the data set for Hunger Games? (Hint: Round mean to the nearest ones.)

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16What is the mean absolute deviation of the data set for Twilight? (Hint: Round mean to the nearest ones.)

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17Which book has more variability in the number of pages per chapter? AHunger Games BTwilight

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18The difference of the means between the two data sets is approximately ______ times the mean absolute deviation for Twilight? (Round your answers to the nearest tenths.)

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Introduction to Probability Return to table of contents

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Probability One way to express probability is to use a fraction. Number of favorable outcomes Total number of possible outcomes Probability of an event =

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Example: What is the probability of flipping a nickel and the nickel landing on heads? Probability Step 1: What are the possible outcomes? Step 2: What is the number of favorable outcomes? Step 3: Put it all together to answer the question. The probability of flipping a nickel and landing on heads is: 1. 2 click

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Probability can be expressed in many forms. For example, the probability of flipping a head can be expressed as: 1 or 50% or 1:2 or.5 2 The probability of randomly selecting a blue marble can be expressed as: 1 or 1:6 or 16.7% or.167 6

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When there is no chance of an event occurring, the probability of the event is zero (0). When it is certain that an event will occur, the probability of the event is one (1) Impossible Unlikely Equally Likely Likely Certain The less likely it is for an event to occur, the probability is closer to 0 (i.e. smaller fraction). The more likely it is for an event to occur, the probability is closer to 1 (i.e. larger fraction).

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Without counting, can you determine if the probability of picking a red marble is lesser or greater than 1/2? It is very likely you will pick a red marble, so the probability is greater than 1/2 (or 50% or 0.5) Click to Reveal What is the probability of picking a red marble? 5656 Click to Reveal Add the probabilities of both events. What is the sum? = 1 6 Click to Reveal

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Note: The sum of all possible outcomes is always equal to 1. There are three choices of jelly beans - grape, cherry and orange. If the probability of getting a grape is 3/10 and the probability of getting cherry is 1/5, what is the probability of getting orange? ? = ? 5 + ? = 1 10 ? The probability of getting an orange jelly bean is 5.

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19Arthur wrote each letter of his name on a separate card and put the cards in a bag. What is the probability of drawing an A from the bag? A0 B1/6 C1/2 D1 A RTH U R Probability = Number of favorable outcomes Total number of possible outcomes Need a hint? Click the box.

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20Arthur wrote each letter of his name on a separate card and put the cards in a bag. What is the probability of drawing an R from the bag? A0 B1/6 C1/3 D1 RTH U R Probability = Number of favorable outcomes Total number of possible outcomes A Need a hint? Click the box.

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21Matt's teacher puts 5 red, 10 black, and 5 green markers in a bag. What is the probability of Matt drawing a red marker? A0 B1/4 C1/10 D10/20 Probability = Number of favorable outcomes Total number of possible outcomes Need a hint? Click the box.

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22What is the probability of rolling a 5 on a fair number cube?

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23What is the probability of rolling a composite number on a fair number cube?

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24What is the probability of rolling a 7 on a fair number cube?

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25You have black, blue, and white t-shirts in your closet. If the probability of picking a black t-shirt is 1/3 and the probability of picking a blue t-shirt is 1/2, what is the probability of picking a white t-shirt?

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26If you enter an online contest 4 times and at the time of drawing its announced there were 100 total entries, what are your chances of winning?

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27Mary chooses an integer at random from 1 to 6. What is the probability that the integer she chooses is a prime number? A B C D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

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28 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. Hat A Hat B Hat C 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. From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011

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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. Hat A Hat B Hat C From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011

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Experimental & Theoretical Probability Return to table of contents

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What are the possible outcomes of each of these?

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number of times the outcome happened number of times experiment was repeated Experimental Probability Flip a coin 5 times and determine the experimental probability of heads. Probability of an event HeadsTails

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Example 1 - Golf A golf course offers a free game to golfers who make a hole-in-one on the last hole. Last week, 24 out of 124 golfers achieved this. Find the experimental probability that a golfer makes a hole-in-one on the last hole. 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. Experimental Probability P(hole-in-one) = # of successes # of trials = = 6 31

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Example 2 - Surveys Of the first 40 visitors through the turnstiles at an amusement park, 8 visitors agreed to participate in a survey being conducted by park employees. Find the experimental probability that an amusement park visitor will participate in the survey. You could expect 1 out of every 5 people to participate in the survey. Or there is a 20% chance of a visitor participating in the survey. Experimental Probability P(participation) = # of successes # of trials = 8 40 = 1 5

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# on Die Picture of Roll Results 1 1 one 2 3 twos 3 1 three 4 0 fours 5 4 fives 6 1 six Sally rolled a die 10 times and the results are shown below. Use this information to answer the following questions.

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29 A B C D What is the experimental probability of rolling a 5? 1/2 5/4 4/5 2/5 # on Die Picture of Roll Results 1 1 one 2 3 twos 3 1 three 4 0 fours 5 4 fives 6 1 six These are the results after 10 rolls of the die

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30 A B C D What is the experimental probability of rolling a 4? 1/2 5/4 4/4 0 # on Die Picture of Roll Results 1 1 one 2 3 twos 3 1 three 4 0 fours 5 4 fives 6 1 six These are the results after 10 rolls of the die

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31 A B C D Based on the experimental probability you found, if you rolled the die 100 times, how many sixes would you expect to get? 6 sixes 10 sixes 12 sixes 60 sixes # on DiePicture of RollResults 1 1 one 2 3 twos 3 1 three 4 0 fours 5 4 fives 6 1 six These are the results after 10 rolls of the die

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32 Mike flipped a coin 15 times and it landed on tails 11 times. What is the experimental probability of landing on heads?

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Theoretical Probability What is the theoretical probability of spinning green? Theoretical Probability

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number of favorable outcomes total number of possible outcomes Theoretical Probability Probability of an event

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Theoretical Probability Example 1 - Marbles Find the probability of randomly choosing a white marble from the marbles shown. There is a 2 in 5 chance of picking a white marble or a 40% possibility. P(white) = # of favorable outcomes # of possible outcomes = =

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Theoretical Probability Example 2 - Marbles Suppose you randomly choose a gray marble. Find the probability of this event. There is a 3 in 10 chance of picking a gray marble or a 30% possibility. P(gray) = # of favorable outcomes # of possible outcomes 3 10 =

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There is a 1 in 2 chance of getting tails when you flip a coin or a 50% possibility. Example 3 - Coins Find the probability of getting tails when you flip a coin. P(tails) = # of favorable outcomes # of possible outcomes 1 2 = Theoretical Probability

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33 A B C D What is the theoretical probability of picking a green marble? 1/8 7/8 1/7 1

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34 A B C D What is the theoretical probability of picking a black marble? 1/8 7/8 1/7 0

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35 A B C D What is the theoretical probability of picking a white marble? 1/8 7/8 1/4 1

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36 A B C D What is the theoretical probability of not picking a white marble? 3/4 7/8 1/7 1

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37What is the theoretical probability of rolling a three? A1/2 B3 C1/6 D1

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38What is the theoretical probability of rolling an odd number? A1/2 B3 C1/6 D5/6

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39What is the theoretical probability of rolling a number less than 5? A2/3 B4 C1/6 D5/6

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40What is the theoretical probability of not rolling a 2? A2/3 B2 C1/6 D5/6

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41Seth tossed a fair coin five times and got five heads. The probability that the next toss will be a tail is A0 B C D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

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42Which inequality represents the probability, x, of any event happening? Ax ≥ 0 B0 < x < 1 Cx < 1 D0 ≤ x ≤ 1 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011

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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.

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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. Compare the experimental probability to the theoretical probability when the experiments for all of the students are considered?

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Word Problems Return to table of contents

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The Marvelous Marble Company produces batches of marbles of 1000 per batch. Each batch contains 317 blue marbles, 576 red marbles, and 107 green marbles. Determine the theoretical probability of selecting each color marble if 1 color is selected by a robotic arm. Number of Outcomes in the Event Total Number of Possible Outcomes Theoretical Probability /1000= 100 = 10.7% 317/1000= 100 = 31.7% 576/1000= 100 = 57.6% = /1000 = 1 100 = 100% 1000/1000 click

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Bob, the manager of the Marvelous Marble Company tells Pete that it is time to add a yellow marble to the batch. In addition, Bob tells Pete to start making the batches in equal proportion so the customer can receive an equal amount of colors in a batch. He tells Pete he needs this taken care of right away. If you were Pete, how would you use theoretical probability to solve this problem? Assume 1000 marbles per batch (red, green, blue and yellow colored marbles) Start with 1000 marbles Divide 1000 into 4 equal parts (equal colors) Each part is equal to 250 marbles Reduce to lowest terms Do you have an explanation of the probability for Bob? Click on black circle to find answer. The customer has a 1 in 4 or 25% chance of picking any color! Click Here

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19 shots made 100 shots attempted = 19% 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. What do you know about percentages to figure out the relationship of goals scored to goals attempted? Experimental Probability =number of times the outcome happened number of times experiment was repeated Please continue on next slide... number of goals number of attempts Erica's Experimental Probability = Move to Reveal click to reveal

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is very close to so she makes about 20% of her shots on goal. Let's estimate the number of goals Erica scored. 1,100 is very close to 1,000. So we will estimate that Erica has about 1,000 attempts About what percent would be a good estimate to use? About how many attempts did Erica take? Erica makes 19% of her shots on goal. Erica takes 1,100 shots on goal. click

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Erica figures she made about 200 of her shots on goal. Erica wants to find 20% of 1,000. Her math looks like this:

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Can you find the actual values that will give you 19%? Challenge

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Example 3 - 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. Based on her experience last year, Lexi can expect 50 out of 60 tulips to bloom. Solve this proportion by looking at it times 5 Experimental Probability 10 bloom 12 total x bloom 60 total = 10 bloom 12 total 50 bloom 60 total =

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Example 4 - Basketball 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. Based on your performance yesterday, you can expect to make 48 free throws out of 75 attempts. Solve this proportion using cross products Experimental Probability 32 75 = 50 x 2400 = 50x 48 = x 32 made 50 attempts x made 75 attempts =

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Number of attempts Number of goals Experimental Probability Now, its your turn. Calculate the experimental probability for the number of goals.

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43Tom was at bat 50 times and hit the ball 10 times. What is the experimental probability for hitting the ball?

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44Tom 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.

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45What is the theoretical probability of randomly selecting a jack from a deck of cards?

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46Mark rolled a 3 on a die for 7 out of 20 rolls. What is the experimental probability for rolling a 3?

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47What is the theoretical probability for rolling a 3 on a die?

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48Some 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?

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49What is the probability of drawing a king or an ace from a standard deck of cards? A2/52 B4/52 C2/13 D8/52

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50What is the probability of drawing a five or a diamond from a standard deck of cards? A4/13 B13/52 C2/13 D16/52

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Fundamental Counting Principle Return to table of contents

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What should I wear today? Buddy has 2 shirts and 3 pairs of pants to choose from. How many different outfits can he make?

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Let's find out how many outfits Buddy can make using a tree diagram.

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Or we could use multiplication to find out how many outfits Buddy could make. 3 2x = 6 pants shirts outfits

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How many different meals can we create using the following menu? SideEntreeDessert Soup Salad French Fries Lasagna Ice Cream Cake Chicken Fajita Burrito Pizza Hamburger

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Create a tree diagram by dragging the items. Side Entree Dessert Soup Salad French Fries Lasagna Ice Cream Cake Chicken Fajita Burrito Pizza Hamburger Lasagna Soup Ice Cream Cake

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Now try to solve the same problem using multiplication. Side Entree Dessert Soup Salad French Fries Lasagna Ice Cream Cake Chicken Fajita Burrito Pizza Hamburger Sides EntreesDesserts Meals x = x

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If you were to pick 4 digits to be your identification number, how many choices are there? Before we begin we must consider if once a number is chosen if it can be repeated. If a digit can repeat its called replacement, because once it chosen it placed back on the list. If a digit cannot repeat it is said to be without replacement, because the number does not back on to the list of choices.

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If you were to pick 4 digits to be your identification number, how many choices are there if there is no replacement? _______ ________ _________ __________ First consider how choices there are for a digit: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 So 10 choices for the first digit. For the second digit there will be only 9 choices left. For the third digit there are only 8 choices left. For the fourth digit there are only 7 choices.

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5,040 combinations Students are given a lock for their gym lockers. Each code requires you to enter 4 single digit numbers. If the numbers cannot be repeated, how many different codes are possible? x = xx Total Possibilities Move to Reveal Answer

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If you were to pick 4 digits to be your identification number, how many choices are there if there is replacement? __________ _________ _________ __________ First consider how choices there are for a digit: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 So 10 choices for the first digit. For the second digit there will be only 10 choices because with replacement there can be repeats. For the third digit there are only 10 choices left. For the fourth digit there are only 10 choices. Using the Counting Principle: (10)(10)(10)(10)= 10,000 combos

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Students are given a lock for their gym lockers. Each code requires you to enter 4 single digit numbers. If the numbers can be repeated, but zero cannot be the first number how many different codes are possible? x = xx Total Possibilities 9,000 combinations Move to Reveal Answer

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7,893,600 combinations This cryptex has a map to treasure buried somewhere in New Jersey inside of it! Each of the 5 columns lists every letter in the alphabet once. What are the total number of codes that can be created if the letters cannot be repeated? Click on lock to reveal answers

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26___ 11,881,376 This cryptex has a map to treasure buried somewhere in New Jersey inside of it! Each of the 5 columns lists every letter in the alphabet once. What is the probability of the codes containing the letters MATH (in that order) as the first 4 letters in the code? (Last letter can be a repeat) Challenge Version Click on lock to reveal answers

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51Robin has 8 blouses, 6 skirts, and 5 scarves. Which expression can be used to calculate the number of different outfits she can choose, if an outfit consists of a blouse, a skirt, and a scarf? A B8 6 5 C8! 6! 5! D 19 C 3 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

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52In a school building, there are 10 doors that can be used to enter the building and 8 stairways to the second floor. How many different routes are there from outside the building to a class on the second floor? A1 B10 C18 D80 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 20

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53 A B C D Joe has 4 different hats, 3 different shirts, and 2 pairs of pants. How many different outfits can Joe make? 9 outfits 14 outfits 24 outfits 12 outfits

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54 A B C D Stacy is trying to find out how many different combinations of license plates there. She lives in New Jersey where there are 3 letters followed by 3 numbers. How many different combinations of license plates are there? 17,576,000 license plates 12,812,904 license plates 729 license plates 17,576 license plates

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55If you wanted to maximize the amount of available license plates and could add an additional letter or number to the existing combination of 3 letters and 3 numbers, would you add a letter or a number? Aletter Bnumber

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56 A B C D Becky and Andy are going on their first date to the movies. Andy wants to buy Becky a snack and drink, but she is taking forever to make a decision. Becky says that there are too many combinations to choose from. If there are 6 different types of drinks and 15 different snacks, how many options does Becky actually have? 45 choices 90 choices 21 choices 42 choices

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Ali is making bracelets for her and her friends out of beads. She figured that each bracelet should be about 10 beads. If she only has blue and green beads, how many different bracelets can she possibly make? 1,024 bracelets 1,000 bracelets 100 bracelets 20 bracelets 57 A B C D

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585 styles of bikes come in 4 colors each, how many different bikes choices are available?

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59If the book store has four levels of algebra books, each level is available in soft back or hardcover, and each comes in three different typefaces, how many options of algebra books are available?

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60How many ways can 3 students be named president, vice president, and secretary if each holds only 1 office?

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61How many ways can a 8-question multiple choice quiz be answered if the there are 4 choices per question?

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62A locker combination system uses three digits from 0 to 9. How many different three- digit combinations with no digit repeated are possible? A30 B504 C720 D1000 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

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63How many different five-digit numbers can be formed from the digits 1, 2, 3, 4, and 5 if each digit is used only once? A120 B60 C24 D20 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

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64All seven-digit telephone numbers in a town begin with 245. How many telephone numbers may be assigned in the town if the last four digits do not begin or end in a zero? From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

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65The telephone company has run out of seven-digit telephone numbers for an area code. To fix this problem, the telephone company will introduce a new area code. Find the number of new seven-digit telephone numbers that will be generated for the new area code if both of the following conditions must be met: The first digit cannot be a zero or a one. The first three digits cannot be the emergency number (911) or the number used for information (411). From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

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Permutations and Combination s Return to table of contents

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How many ways can the following animals be arranged? There are two methods to solve this problem: Method 1: List all the possible groupings Method 2: Use the permutation.

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Method 1: List all possible groupings. There are 24 arrangements of 4 animals in 4 positions.

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4 3 2 1 = 24 There are 4 choices for the first position. There are 3 choices for the second position. There are 2 choices for the third position. There is 1 choice for the fourth position. There are 24 arrangements of 4 animals in 4 positions. The expression 4 3 2 1 can be written as 4!, which is read as "4 factorial." Method 2: Use the permutation. A permutation is an arrangement of n objects in which order is important.

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66 What is the value of 5! ?

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67 How many ways can the letters in FROG be arranged?

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68 In how many ways can a police officer, fireman and a first aid responder enter a room single file? A3 B3! C6 D6! E1

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69 In how many ways can four race cars finish a race that has no ties? A4 B4! C24 D24! E12

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70How many ways can the letters the word HOUSE be arranged?

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71How many ways can 6 books be arranged on a shelf?

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How many ways can the letters in the word DEER be rearranged? There are 2 E's! So DEER and DEER are consider to be the same combo. Since there are 2 repeated letters calculate the combos using the Counting Principle and the divide by 2. (4)(3)(2)(1) = 12 ways 2

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72 In how many ways can the letters in JERSEY be arranged?

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73How many different three-letter arrangements can be formed using the letters in the word ABSOLUTE if each letter is used only once? A56 B112 C168 D336 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

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Permutation Formula Key concept: an arrangement of n objects in which order is important is a permutation. A race is an example of a situation where order is important. Can you name other examples where order is important? ____________________________________________________ The number of permutations of n objects taken r at a time can be written as n P r, where n P r =

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If 5 cars were in a race and prizes were awarded for first, second and third, this is the number of possible ways for the prizes to be awarded. 5 P 3 =

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Note Always remember that: 0! = 1 1! = 1 If 5 cars were in a race and prizes were awarded for each racer, the number of possible ways for the prizes to be awarded would be 5 P 5 = 5! (5-5)! = 5! 0! = =

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74 Find the value of 6P26P2

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75 Find the value of 4P14P1

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76 Find the value of 6P66P6

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Twenty young ladies entered a beauty contest. Prizes will be awarded for first, second and third place. How many different ways can the first, second and third place prizes be awarded? 20 P 3 = = 20! = 20 19 18 17! 17! 17! = 6840

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Find the number of permutations of 4 objects taken 3 at a time. How many 4-digit numbers can you make using each of the digits 1, 2, 3, and 4 exactly once? 4 P 4 = 4! = 4 3 2 1 = 24 0! 1

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7710 cars are in a race. How many ways can prizes be awarded for first, second and third place?

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78How many ways can four out of seven books be arranged on a shelf?

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79You are taking 7 classes, three before lunch. How many possible arrangements are there for morning classes?

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80The teacher is going to select a president and vice-president from the 24 students in class. How many possible arrangements are there for president and vice-president?

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Combinations A combination is a selection of objects when order is not important. Example: A combination pizza, since it does not matter in which order the toppings were placed. Can you think of other examples when order does not matter?

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81 You must read 5 of the 10 books on the summer reading list. This is an example of a _________ ACombination BPermutation

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82 You must fit 5 of the 10 books on the shelf. How many different ways are there to place them on the shelf? This is an example of a ____________ ACombination BPermutation

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83 10 people are in a room. How many different pairs can be made? This is an example of a ____________ ACombination BPermutation

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84 10 people are about to leave a room. How many different ways can they walk out of the room? This is an example of a ____________ ACombination BPermutation

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85 You have 100 relatives and can only invite 50 to your 16th birthday party. The possibilities of who can be invited is an example of a ____________ ACombination BPermutation

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Combinations ________________________________________________ To find the number of combinations of n objects taken r at a time, divide the number of permutations of n objects taken r at a time by r! n C r = n P r r! _________________________________________________

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There are 7 pizza toppings and you are choosing four of them for your pizza. How many different pizzas are possible to create? The order in which you choose the toppings is not important, so this is a combination. To find the number of different ways to choose 4 toppings from 7, find 7 C 4. 7 C 4 = 7 P 4 = 7 6 5 4 = 35 4! 4 3 2 1

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86 Find the number of combinations. 5 C 2

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87 There are 40 students in the computer club. Five of these students will be selected to compete in the ALL STAR competition. How many different groups of five students can be chosen?

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88There are 45 flowers in the shop. How many different arrangements containing 10 flowers can be created?

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89Eight people enter the chess tournament. How many different pairings are possible?

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90Mary can select 3 of 5 shirts to pack for the trip. How many different groupings are possible?

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91How many different three-member teams can be selected from a group of seven students? A1 B35 C210 D5040

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Probability of Compound Events Return to table of contents

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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)

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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) 4 4 = _1_

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When the outcome of one event affects the outcome of another event, the two events are dependent. Use formula: Probability (A & B) = Probability(A) Probability(B given A) Select a card from a deck of cards, do not 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? Dependent Example P(6 and a king) = P(6) P(king given a six has been selected) 4 4 =

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92 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?

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93 A lottery machine generates numbers randomly. Two numbers between 1 and 9 are generated. What is the probability that both numbers are 5?

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94 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?

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95 What is the probability that the first two cards drawn from a full deck are both hearts? (without replacement)

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96 A spinner containing 5 colors: red, blue, yellow, white and green is spun and a die, numbered 1 thru 6, is rolled. What is the probability of spinning green and rolling a two?

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97 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?

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98At a school fair, the spinner represented in the accompanying diagram is spun twice. A B C D RG B What is the probability that it will land in section G the first time and then in section B the second time? From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

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99 A student council has seven officers, of which five are girls and two are boys. If two officers are chosen at random to attend a meeting with the principal, what is the probability that the first officer chosen is a girl and the second is a boy? A B C D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

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100The probability that it will snow on Sunday is. The probability that it will snow on both Sunday and Monday is. What is the probability that it will snow on Monday, if it snowed on Sunday? A B2 C D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

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Probabilities of Mutually Exclusive & Overlapping Events Return to table of contents

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Events are mutually exclusive or disjoint if they have no outcomes in common. Example: Event A: Roll a 3 Event B: Roll an even number Event A 3 Event B 2 4 6

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Overlapping Events are events that have one or more outcomes in common Example Event A: Roll an even number Event B: Roll a number greater than 3 Event A 2 Event B

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101Are the events mutually exclusive? Event A: Selecting an Ace Event B: Selecting a red card A Yes B No

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102Are the events mutually exclusive? Event A: Rolling a prime number Event B: Rolling an even number A Yes B No

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103Are the events mutually exclusive? Event A: Rolling a number less than 4 Event B: Rolling an even number A Yes B No

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104Are the events mutually exclusive? Event A: Selecting a piece of fruit Event B: Selecting an apple A Yes B No

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105Are the events mutually exclusive? Event A: Roll a multiple of 3 Event B: Roll a divisor of 19 A Yes B No

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106Are the events mutually exclusive? Randomly select a football card Event A: Select a Philadelphia Eagle Event B: Select a starting quarterback A Yes B No

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107Are the events mutually exclusive? Event A: The Yankees won the World Series Event B: The Mets won the National League Pennant A Yes B No

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Formula probability of two mutually exclusive events P(A or B) = P(A) + P(B) Take notes!

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What is the probability of drawing a 5 or an Ace from a standard deck of cards? There are 52 outcomes for the standard deck. 4 of these cards are 5s and 4 are Aces. There is not a card that is both a 5 and an A. So... P(5 or A) = P(5) + P(A) = reduce 2 13 Check your answer by pulling down the screen.

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Find the probability if you if you roll a pair of number cubes and the numbers showing are the same or that the sum is 11. P(numbers =) + P(sum is 11) = 8 or Click to reveal answer

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A bag contains the following candy bars: 3 Snickers 4 Mounds 2 Almond Joy 1 Reese's Peanut Butter Cup You randomly draw a candy bar from the bag. What is the probability that you select a Snickers or a Mounds bar? Are the events mutually exclusive? Find the probability that you select a Snickers bar Find the probability that you select a Mounds bar Find the probability that you select a Snickers or a Mounds bar

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108In a room of 100 people, 40 like Coke, 30 like Pepsi, 10 like Dr. Pepper, and 20 drink only water. If a person is randomly selected, what is the probability that the person likes Coke or Pepsi?

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109In a school election, Bob received 25% of the vote, Cara received 40% of the vote, and Sam received 35% of the vote. If a person is randomly selected, what is the probability that the person voted for Bob or Cara?

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110 A die is rolled twice. What is the probability that a 4 or an odd number is rolled?

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111Sal has a small bag of candy containing three green candies and two red candies. While waiting for the bus, he ate two candies out of the bag, one after another, without looking. What is the probability that both candies were the same color?

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112 Events A and B are disjoint. Find P(A or B). P(A) = P(B) =

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113Events A and B are disjoint. Find P(A or B). P(A) = P(B) =

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If the situation is 2 events CAN occur at the same time, then these are NOT mutually exclusive events. Think about this... What's the problem with this situation... What is the probability of selecting a black card or a 7? P(black or 7)

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Formula Probability of two events which are NOT mutually exclusive P(A or B) = P(A) + P(B) - P(A and B) Take notes!

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What is the probability of selecting a black card or a 7? P(black or 7) P(black or 7) = P(black) + P(7) - P(black and 7) P(black or 7) = = 28 = 7_

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Of the 300 students at Jersey Devil Middle School, 121 are girls, 16 students play softball, 29 students are on the lacrosse team and, 25 are girls on the lacrosse team. Find the probability that a student chosen at random is a girl or is on the lacrosse team. 300 total students girls lacrosse

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Of the 300 students at Jersey Devil Middle School, 121 are girls, 16 students play softball, 29 students are on the lacrosse team and, 25 are girls on the lacrosse team. Find the probability that a student chosen at random is a girl or is on the lacrosse team. P(girl or lacrosse) = P(girl) + P(lacrosse) - P(girl and lacrosse) =0.416 Now do the math! click to reveal

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114 A B C D In a special deck of cards each card has exactly one different number from 1-19 (inclusive) on it. Which gives the probability of drawing a card with an odd number or a multiple of 3 on it? P(odd) + P(multiple of 3) P(odd) x P (multiple of 3) - P(odd and multiple of 3) P(odd) x P(multiple of 3) P(odd) + P (multiple of 3) - P(odd and multiple of 3)

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115Events A and B are overlapping. Find P(A or B). P(A) = P(B) = P(A and B) =

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116Events A and B are overlapping. Find P(A or B). P(A) = P(B) = P(A and B) =

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117What is the probability of rolling a number less than two or an odd number?

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118What is the probability of rolling a number that is not even or that is not a multiple of 3?

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Complementary Events Return to table of contents

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Complementary Events Two events are complementary events if they are mutually exclusive and one event or the other must occur. The sum of the probabilities of complementary events is always 1. P(A) + P(not A) = 1 Example: The forecast calls for a 30% chance of rain. What is the probability that it will not rain? P(rain) + P(not rain) = ? = 1 P(not rain) =.7

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119Given P(A), find P(not A). P(A) = 52% P(not A) = ______ %

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120 Given P(A), find P(not A). P(A) = P(not A) = ______

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121The spinner below is divided into eight equal regions and is spun once. What is the probability of not getting red? A B C D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, Green Yellow Red Blue White Red Purple

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122The faces of a cube are numbered from 1 to 6. What is the probability of not rolling a 5 on a single toss of this cube? A B C D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

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