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www.njctl.org 2013-05-31

7th Grade Math Probability

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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. Pull Pull

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1 Food services at your school wants to increase the number

• f 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 Pull Pull

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2 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 No Pull Pull

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3 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 No Pull Pull

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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

• f 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 173.57 = 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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4

860 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?

Pull

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5

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

Pull

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6 You are an inspector. You find 3 faulty bulbs out of 50.

Estimate the number of faulty bulbs in a lot of 2,000. Pull Pull

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7 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? Pull Pull

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8

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

high-top sneakers. Explain your reasoning.

Number of Students Low-top sneakers 12 High-top sneakers 7 Sandals 3 Boots 6 Shoes

Pull Pull

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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 Sample Hamburgers

Tacos Pizza Total

#1 12

14 74 100

#2 12

11 77 100 · 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 Sample

Cars SUVs Mini Vans Motorcycles Total

#1

37 43 12 8 100

#2

33 46 11 10 100 Pull Pull

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Note to Teacher

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 Regular BBQ Cheddar #1 8 10 7 #2 8 11 6

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9

How many students participated in each survey?

Pull Pull

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10 According to the two random samples, which flavor potato

chip should the student council purchase the most of? A Regular B BBQ C Cheddar Pull Pull

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11

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

Pull Pull

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Return to Table of Contents

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Measure of Center - Vocabulary Review Mean (Average) - The sum of the data values divided by the number of items Median - The middle data value when the values are written in numerical order Mode - The data value that occurs the most often

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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

Example from KATM.org

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65 70 75 80 85 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

Height of Soccer Players (inches)

65 70 75 80 85 x x x x x x x x x x x x x x x x

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.

Slide 37 / 231 Chart of just heights Slide 38 / 231

Since the teams have different numbers of players, Victor wants to determine the mean height for each team. The difference between the means (mh2 - mh1) is about 8 inches. The mean height of a soccer player is about 72 inches. The mean height of a basketball player is about 80 inches.

Sum of Soccer Player's Height Total # of Soccer Players mh1 = Sum of Basketball Player's Height Total # of Basketball Players mh2 = = 2090 in = 72.07 in 29 = 1276 in = 79.75 in 16

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The mean absolute deviation can tell us more about the variability of data within a set. Victor decides to calculate that next.

mad = sum of absolute deviation values* # of players * absolute deviation value = distance between mean height and individual height value

Example with Soccer players (1st data value): 65 in - 72 in = 7 in Example with Soccer players (1st data value): 65 in - 72 in = 7 in

* absolute deviation value = distance between mean height and individual height value mad = sum of absolute deviation values* # of players

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The mean absolute deviation is 2.14 inches for a soccer player. This means that the average soccer player varies 2.14 inches in height from 72 inches. The mean absolute deviation is 2.5 inches for a soccer player. This means that the average basketball player varies 2.5 inches in height from 80 inches. The mean absolute deviations for both teams are very close. This means the variabilities are similar.

mad1 = sum of absolute deviation values # of soccer players mad2 = sum of absolute deviation values # of basketball players

= 62 in = 2.14 in 29 = 40 in = 2.5 in 16

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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 79.75 - 72.07 = 7.68 Divide the difference by the mean absolute deviations of each data set. 7.68 ÷ 2.14 = 3.59 7.68 ÷ 2.5 = 3.07 The difference of the means (7.68) is approximately 3 times greater than 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 10 15 20 25 30

x x x x x x x x x x x x x x x x x x x x x x x x x x x

10 15 20 25 30

x x x x x x x x x x x x x x x x x x x x x x

Pages per Chapter in Twilight

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12

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

Pull Pull

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13

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

Pull Pull

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14

What is the difference of the means?

Pull Pull

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15

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

Pull Pull

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16

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

Pull Pull

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17 Which book has more variability in the number of pages per

chapter? A Hunger Games B Twilight Pull Pull

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18

The 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.)

Pull Pull

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Introduction to Probability

Click to go 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

• f an event

= Slide 53 / 231

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?

Pull Pull

Step 3: Put it all together to answer the question. The probability of flipping a nickel and landing on heads is: 1 . 2

Pull Pull

click 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). 1 4 1 2 3 4 1 I m p

• s

s i b l e Unlikely Equally Likely Likely C e r t a i n 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? 5 6

Click to Reveal

Add the probabilities of both events. What is the sum? 1 + 5 = 1 6 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? 3 + 1 + ? = 1 10 5 ? 5 + ? = 1 10 ? The probability of getting an orange jelly bean is 5 . 10

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19

Arthur 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? A B 1/6 C 1/2 D 1

A R T H U R

Probability = Number of favorable outcomes Total number of possible outcomes

Pull Pull

Need a hint? Click the box.

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20

Arthur 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? A B 1/6 C 1/3 D 1

A R T H U R

Probability = Number of favorable outcomes Total number of possible outcomes

A

Pull Pull

Need a hint? Click the box.

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21

Matt's teacher puts 5 red, 10 black, and 5 green markers in a bag. What is the probability of Matt drawing a red marker?

A 0 B

1/4

C

1/10

D

10/20

Pull Pull

Probability = Number of favorable outcomes Total number of possible outcomes

Need a hint? Click the box.

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22

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

Pull Pull

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23

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

Pull Pull

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24

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

Pull Pull

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25

You 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?

Pull Pull

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26

If 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?

Pull Pull

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27

Mary 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 www.nysedregents.org/IntegratedAlgebra; accessed 17,

June, 2011.

Pull Pull

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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 www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011

Pull Pull

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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 www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011

Pull Pull

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Experimental & Theoretical Probability

Click to go to Table of Contents

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Click on an object. What is the outcome?

O u t c

• m

e

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number of times the outcome happened

number of times experiment was repeated

Experimental Probability

Flip the coin 5 times and determine the experimental probability of heads.

Answers

Probability

• f an event

Heads Tails

Pull Pull

The ratio of the number of times an event occurs to the total number of times that the activity is performed.

Experimental Probability

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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 = 24 124 = 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 Pull Pull

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30

A B C D

What is the experimental probability of rolling a 4? 1/2 5/4 4/4 # 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 Die Picture of Roll Results 1 1 one 2 3 twos 3 1 three 4 fours 5 4 fives 6 1 six

These are the results after 10 rolls of the die Pull Pull

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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? Pull Pull

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Theoretical Probability

Pull the tabs for definitions.

What is the theoretical probability of spinning green?

Probability Probability Probability

A n s w e r Theoretical Probability Equally Likely

Fair

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number of favorable outcomes total number of possible outcomes

Theoretical Probability

Probability

• f 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 4 2 10 5 = =

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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. Theoretical Probability Example 3 - Coins Find the probability of getting tails when you flip a coin. P(tails) = # of favorable outcomes # of possible outcomes 1 2 =

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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

R R G W W Y Y B Pull Pull

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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

R R G W W Y Y B Pull Pull

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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

Pull Pull R R G W W Y Y B

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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

Pull Pull R R G W W Y Y B

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37 What is the theoretical probability of rolling a three?

A 1/2 B 3 C 1/6 D 1 Pull Pull

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38 What is the theoretical probability of rolling an odd number?

A 1/2 B 3 C 1/6 D 5/6 Pull Pull

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39 What is the theoretical probability of rolling a number less

than 5? A 2/3 B 4 C 1/6 D 5/6 Pull Pull

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40 What is the theoretical probability of not rolling a 2?

A 2/3 B 2 C 1/6 D 5/6 Pull Pull

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41

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 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.

Pull Pull

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42

Which inequality represents the probability, x, of any event happening? A x ≥ 0 B 0 < x < 1 C x < 1 D 0 ≤ x ≤ 1

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

Pull Pull

Slide 94 / 231 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

Click to go 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

576 317 107 1000 1000 1000 107/1000=0.107 0.107 100= 10.7% 317/1000=0.317 0.317 100= 31.7% 576/1000=0.576 0.576 100= 57.6% 107+317+576=1000 1000/1000=1 100= 100% 1000/1000

click click 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

• r 25%

chance of picking any color!

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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 Move to Reveal

click to reveal

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19 100 20 100 is very close to so she makes about 20%

• f 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 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:

click to reveal

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

Challenge

Hint Answer

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Example 3 - Gardening Last year, Lexi planted 12 tulip bulbs, but only 10

• f 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

• ut 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 100 1000 500 2000 30 600 150

1600

Now, its your turn. Calculate the experimental probability for the number of goals.

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43 Tom was at bat 50 times and hit the ball 10 times. What is

the experimental probability for hitting the ball? Pull Pull

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44 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. Pull Pull

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45 What is the theoretical probability of randomly selecting a

jack from a deck of cards? Pull Pull

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

experimental probability for rolling a 3? Pull Pull

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

Pull Pull

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48 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? Pull Pull

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49 What is the probability of drawing a king or an ace from a

standard deck of cards? A 2/52 B 4/52 C 2/13 D 8/52 Pull Pull

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50 What is the probability of drawing a five or a diamond from

a standard deck of cards? A 4/13 B 13/52 C 2/13 D 16/52 Pull Pull

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Fundamental Counting Principle

Click to go to Table of Contents

Slide 115 / 231 What should I wear today? Buddy has 2 shirts and 3 pairs

• f pants to choose from. How

many different outfits can he make? Slide 116 / 231

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

Pull Pull

To make a tree diagram, match each pair of pants with each shirt. Buddy can make 6 outfits!

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Or we could use multiplication to find

• ut how many outfits Buddy could

make. 3 2 x = 6 pants shirts

• utfits

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

Slide 119 / 231 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 Entrees Desserts 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? 1 2 3 4 5 6 7 8 9 x = x x 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7

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

Slide 125 / 231

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? 1 2 3 4 5 6 7 8 9 x = x x 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

Total Possibilities 9,000 combinations

M

• v

e t

• R

e v e a l A n s w e r

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7,893,600 combinations This cryptex has a map to treasure buried somewhere in New Jersey inside of it! Each

• f 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?

Hint 1 Hint 2

Hint 3

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

• f 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)

Hint 1 Hint 2

Hint 3 Challenge Version Click on lock to reveal answers

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51

Robin 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

• f a blouse, a skirt, and a scarf?

A 8 + 6 + 5 B 8 • 6 • 5 C 8! 6! 5! D

19C3

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.

Pull Pull

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52

In 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? A 1 B 10 C 18 D 80

From the New York State Education Department. Office of Assessment Policy, Development and

• Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17,

June, 20

Pull Pull

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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 Pull Pull

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54

A B C D Stacy is trying to find out how many different combinations

• f 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 Pull Pull

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55 If 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? A letter B number Pull Pull

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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 Pull Pull

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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

• nly 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 Pull Pull

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58

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

Pull Pull

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59

If 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?

Pull Pull

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60

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

Pull Pull

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61

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

Pull Pull

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62

A locker combination system uses three digits from 0 to 9. How many different three-digit combinations with no digit repeated are possible? A 30 B 504 C 720 D 1000

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.

Pull Pull

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63

How many different five-digit numbers can be formed from the digits 1, 2, 3, 4, and 5 if each digit is used only once? A 120 B 60 C 24 D 20

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.

Pull Pull

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64

All 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 www.nysedregents.org/IntegratedAlgebra; accessed 17,

June, 2011.

Pull Pull

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65 The 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 www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Pull Pull

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Permutations and Combinations

Click to go to Table of Contents

Slide 144 / 231

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! ?

Pull Pull

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

Pull Pull

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68 In how many ways can a police officer, fireman and a first

aid responder enter a room single file? A 3 B 3! C 6 D 6! E 1 Pull Pull

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69 In how many ways can four race cars finish a race that has

no ties? A 4 B 4! C 24 D 24! E 12 Pull Pull

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70 How many ways can the letters the word HOUSE be

arranged? Pull Pull

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

Pull Pull

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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?

Pull Pull

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73 How many different three-letter arrangements can be formed

using the letters in the word ABSOLUTE if each letter is used

• nly once?

A 56 B 112 C 168 D 336

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.

Pull Pull

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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

nPr, where nPr =

Slide 157 / 231

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.

5P3 =

Slide 158 / 231 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

5P5 =

5! (5-5)! = 5! 0! = 120 1 120 =

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74 Find the value of

6P2

Pull Pull

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75 Find the value of 4P1

Pull Pull

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76 Find the value of 6P6

Pull Pull

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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?

20P3 =

Slide 163 / 231

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 P4 =

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77 10 cars are in a race. How many ways can prizes be awarded

for first, second and third place? Pull Pull

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78 How many ways can four out of seven books be arranged

• n a shelf?

Pull Pull

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79 You are taking 7 classes, three before lunch. How many

possible arrangements are there for morning classes? Pull Pull

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80 The 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? Pull Pull

Slide 168 / 231

Combinations

A combination is a selection of objects when order is not important. Example: A combination pizza, since it does not matter in which

• rder 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 ____________

A Combination B Permutation Pull Pull

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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 ____________ A Combination B Permutation Pull Pull

Slide 171 / 231

83 10 people are in a room. How many different pairs can be

made? This is an example of a ____________ A Combination B Permutation Pull Pull

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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 ____________ A Combination B Permutation Pull Pull

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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 ____________ A Combination B Permutation Pull Pull

Slide 174 / 231

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! ____________________________________________________________ ____

Slide 175 / 231

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

Pull Pull

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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? Pull Pull

Slide 178 / 231

88 There are 45 flowers in the shop. How many different

arrangements containing 10 flowers can be created? Pull Pull

Slide 179 / 231

89 Eight people enter the chess tournament. How many

different pairings are possible? Pull Pull

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90 Mary can select 3 of 5 shirts to pack for the trip. How many

different groupings are possible? Pull Pull

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91

How many different three-member teams can be selected from a group of seven students? A 1 B 35 C 210 D 5040

Pull Pull

Slide 182 / 231

Probability of Compound Events

Click to go to Table of Contents

Slide 183 / 231

Probability of Compound Events

First - decide if the two events are independent or dependent. When the outcome of one event does not affect the

• utcome of another event, the two events are

independent. Use formula: Probability (A and B) = Probability (A) Probability (B)

Slide 184 / 231

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_ 52 52 169

Slide 185 / 231

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 = 4 52 51 663

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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? Pull Pull

Slide 187 / 231

93

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

Slide 188 / 231

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? Pull Pull

Slide 189 / 231

95

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

Slide 190 / 231

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? Pull Pull

Slide 191 / 231

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? Pull Pull

Slide 192 / 231

98 At a school fair, the spinner represented in the accompanying diagram is

spun twice.

A B C D

R G 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 www.nysedregents.org/IntegratedAlgebra; accessed 17,

June, 2011.

Pull Pull

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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 www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Pull Pull

Slide 194 / 231

100 The 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 B

2

C 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.

Pull Pull

Slide 195 / 231

Probabilities of Mutually Exclusive & Overlapping Events

Click to go to Table of Contents

Slide 196 / 231

Events are mutually exclusive or disjoint if they have no

• utcomes in common.

Example: Event A: Roll a 3 Event B: Roll an even number Event A 3 Event B 2 4 6

Slide 197 / 231

Overlapping Events are events that have one or more

• utcomes in common

Example Event A: Roll an even number Event B: Roll a number greater than 3 Event A 2 Event B 5 4 6

Slide 198 / 231

101 Are the events mutually exclusive?

Event A: Selecting an Ace Event B: Selecting a red card Yes No Pull Pull

Slide 199 / 231

102 Are the events mutually exclusive?

Event A: Rolling a prime number Event B: Rolling an even number Yes No Pull Pull

Slide 200 / 231

103 Are the events mutually exclusive?

Event A: Rolling a number less than 4 Event B: Rolling an even number Yes No Pull Pull

Slide 201 / 231

104 Are the events mutually exclusive?

Event A: Selecting a piece of fruit Event B: Selecting an apple Yes No Pull Pull

Slide 202 / 231

105 Are the events mutually exclusive?

Event A: Roll a multiple of 3 Event B: Roll a divisor of 19 Yes No Pull Pull

Slide 203 / 231

106 Are the events mutually exclusive?

Randomly select a football card Event A: Select a Philadelphia Eagle Event B: Select a starting quarterback Yes No Pull Pull

Slide 204 / 231

107 Are the events mutually exclusive?

Event A: The Yankees won the World Series Event B: The Mets won the National League Pennant Yes No Pull Pull

Slide 205 / 231

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

T a k e n

• t

e s !

Pull Pull

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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) 4 + 4 = 8 52 52 52 reduce 2 13 Check your answer by pulling down the screen.

Slide 207 / 231

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)

Note to Teacher

Slide 208 / 231

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

Pull Pull

Slide 209 / 231

108 In 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

• r Pepsi?

Pull Pull

Slide 210 / 231

109 In 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? Pull Pull

Slide 211 / 231

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

• dd number is rolled?

Pull Pull

Slide 212 / 231

111 Sal 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? Pull Pull

Slide 213 / 231

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

P(A) = P(B) = Pull Pull

Slide 214 / 231

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

P(A) = P(B) = Pull Pull

Slide 215 / 231 If the situation is 2 events CAN

• ccur at the same time, then

these are NOT mutually exclusive events.

T h i n k a b

• u

t t h i s . . .

What's the problem with this situation... What is the probability of selecting a black card or a 7? P(black or 7)

Slide 216 / 231

Formula

probability of two events which are NOT mutually exclusive P(A or B) = P(A) + P(B) - P(A and B)

T a k e n

• t

e s !

Slide 217 / 231

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) = 26 + 4 - 2 = 28 = 7 52 52 52 52 13

Slide 218 / 231

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

Pull Pull

Slide 219 / 231

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) 121 300 29 300 25 300 +

• 125

300 = 0.416

Now do the math!

click to reveal click to reveal

Slide 220 / 231

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) Pull Pull

Slide 221 / 231

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

P(A) = P(B) = P(A and B) = Pull Pull

Slide 222 / 231

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

P(A) = P(B) = P(A and B) = Pull Pull

Slide 223 / 231

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

an odd number? Pull Pull

Slide 224 / 231

118 What is the probability of rolling a number that is not even

• r that is not a multiple of 3?

Pull Pull

Slide 225 / 231

Complementary Events

Click to go to Table of Contents

Slide 226 / 231

Complementary Events

Two events are complementary events if they are mutually exclusive and one event or the other must

• ccur. 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 .3 + ? = 1 P(not rain) = .7

Slide 227 / 231

119 Given P(A), find P(not A).

P(A) = 52% P(not A) = ______ % Pull Pull

Slide 228 / 231

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

P(A) = P(not A) = ______ Pull Pull

Slide 229 / 231

121 The spinner below is divided into eight equal regions and is

spun once. What is the probability of not getting red? A B C D

Green Yellow Red Blue Red Red Red Red Blue Purple White

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.

Pull Pull

Slide 230 / 231

122 The 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 www.nysedregents.org/IntegratedAlgebra; accessed 17,

June, 2011.

Pull Pull

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