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www.njctl.org 2012-05-05

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 This notebook appears in both Pre-Algebra and Algebra.

Click on a topic to go to that section.

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Introduction to Probability

Click to go to Table of Contents

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Probability

· Another way to express probability is to use a fraction.

Number of favorable outcomes Total number of possible outcomes Probability

• f an event

= Slide 6 / 176

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

This number becomes your numerator. P = 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

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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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1 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

Need a hint? Move the box.

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2 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

Need a hint? Move the box.

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3 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 B 1/4 C 1/10 D 10/20

Probability = Number of favorable outcomes Total number of possible outcomes

Need a hint? Move the box.

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

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

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

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7 If you have 3 black t-shirts and 4 blue t-shirts, what is the probability of picking a black t-shirt without looking?

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8 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?

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9 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.

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10 Each of the hats shown below has colored marbles placed inside. Hat A contains five green marbles and four red marbles. Hat B contains six blue marbles and five red marbles. Hat C contains five green marbles and five blue marbles.

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

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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

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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? Outcomes are the different results that can occur.

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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11 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

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12 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

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13 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

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14 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

Pull the tabs for definitions.

What is the theoretical probability of spinning green?

Probability Probability Probability

Answer Theoretical Probability E q u a l l y L i k e l y

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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15 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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16 A B C D

What is the theoretical probability of picking a black marble? 1/8 7/8 1/7 1

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17 A B C D

What is the theoretical probability of picking a white marble? 1/8 7/8 1/7 1

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18 A B C D

What is the theoretical probability of not picking a white marble? 1/8 7/8 1/7 1

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19 What is the theoretical probability of rolling a three? A 1/2 B 3 C 1/6 D 1

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20 What is the theoretical probability of rolling an odd number? A 1/2 B 3 C 1/6 D 5/6

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21 What is the theoretical probability of rolling a number less than 5? A 2/3 B 4 C 1/6 D 5/6

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22 What is the theoretical probability of not rolling a 2? A 2/3 B 2 C 1/6 D 5/6

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23 Seth tossed a fair coin five times and got five heads. The probability that the next toss will be a tail is A B C 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.

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24 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

Slide 44 / 176 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

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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

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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.

(please click on the boxes to see if you are correct)

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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

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 .30 or 30% .60 or 60% .30 or 30% .80 or 80%

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

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

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26 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.

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

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28 Mark 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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29 What is the theoretical probability for rolling a 3 on a die?

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30 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?

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Fundamental Counting Principle

Click to go to Table of Contents

Slide 63 / 176 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 64 / 176

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 67 / 176 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

Slide 68 / 176 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

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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? 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

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

• 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?

H i n t 1 Hint 2

Hint 3

Click on lock to reveal answers

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25 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)

H i n t 1 Hint 2

Hint 3 C h a l l e n g e V e r s i

• n

Click on lock to reveal answers

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31 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.

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32 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

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33 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

T

• e

n l a r g e c a l c u l a t

• r

, p u l l t h e b

• t

t

• m

r i g h t corner.

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34 A B C D

T

• e

n l a r g e c a l c u l a t

• r

, p u l l t h e b

• t

t

• m

r i g h t corner.

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

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35 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

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36 A B C D

T

• e

n l a r g e c a l c u l a t

• r

, p u l l t h e b

• t

t

• m

r i g h t corner.

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

• nly has blue and green beads, how many different bracelets

can she possibly make? 1,024 bracelets 1,000 bracelets 100 bracelets 20 bracelets 37 A B C D

T

• e

n l a r g e c a l c u l a t

• r

, p u l l t h e b

• t

t

• m

r i g h t corner.

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38 5 styles of bikes come in 4 colors each, how many different bikes choices are available?

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39 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?

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

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41 How many ways can a 4-question multiple choice quiz be answered if the there are 5 choices per question?

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42 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.

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43 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.

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44 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.

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45 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.

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Permutations and Combinations

Click to go 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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46 What is the value of 5! ?

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

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48 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

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49 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

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

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51 How many ways can 4 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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52 In how many ways can the letters in JERSEY be arranged?

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53 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.

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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 = 5P3 =

= = = 60 _________________________________________________________ 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.

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54 Find the value of 6P2

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55 Find the value of 4P1

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56 Find the value of 5P5 Hint: 0! = 1

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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 =

= 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 P4 = 4! = 4 3 2 1 = 24

0! 1

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57 10 cars are in a race. How many ways can prizes be awarded for first, second and third place?

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

• n a shelf?

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

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60 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?

Slide 114 / 176 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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61 You must read 5 of the 10 books on the summer reading

• list. This is an example of a ____________

A Combination B Permutation

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62 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

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63 10 people are in a room. How many different pairs can be made? This is an example of a ____________ A Combination B Permutation

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64 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

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65 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

Slide 120 / 176 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! ____________________________________________________________ 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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66 Find the number of combinations.

5 C 2

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67 There are 40 students in the computer club. Five of these students will be selected to compete in the ALL STAR

• competition. ow many different groups of five students can

be chosen?

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68 There are 75 flowers in the shop. How many different arrangements containing 10 flowers can be created?

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

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

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

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Probability of Compound Events

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Slide 128 / 176

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)

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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_ 52 52 169

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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 = 4 52 51 663

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72 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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73 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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74 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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75 What is the probability that the first two cards drawn from a full deck are both hearts? (without replacement)

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76 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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77 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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78 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.

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79 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.

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80 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.

Slide 140 / 176 Probabilities of Mutually Exclusive & Overlapping Events

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Slide 141 / 176

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

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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

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

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

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

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

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

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

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

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

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

Take notes!

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.

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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 are the same) + P(sum is 11)

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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

Pull Pull

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89 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?

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90 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?

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

• dd number is rolled?

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92 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?

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

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

Slide 161 / 176 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... Find 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) = 26 + 4 - 2 = 28 = 7_ 52 52 52 52 13

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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

Pull Pull

Slide 165 / 176

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!

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95 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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96 Events A and B are overlapping. Find P(A or B). P(A) = P(B) = P(A and B) =

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

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

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

• r that is not a multiple of 3?

Slide 171 / 176 Complementary Events

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Slide 172 / 176

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

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

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

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102 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.

Slide 176 / 176

103 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.