New Jersey Center for Teaching and Learning Slide 1 / 176 Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org Slide 2 / 176 Probability 2012-05-05 www.njctl.org Introduction to Probability Experimental and Theoretical Word Problems PROBABILITY Fundamental Counting Principle Permutations and Combinations Probability of Compound Events Probabilities of Mutually Exclusive and Overlapping Events Complementary Events Click on a topic to go to that section. Slide 3 / 176 This notebook appears in both Pre-Algebra and Algebra.
Slide 4 / 176 Introduction to Probability Click to go to Table of Contents Probability Slide 5 / 176 Another way to express probability is to use a fraction. Probability of an event = Number of favorable outcomes Total number of possible outcomes Probability Slide 6 / 176 Example: What is the probability of flipping a nickel and the nickel landing on heads? Pull Step 1: What are the possible outcomes? Step 2: What is the number of favorable outcomes? 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
Probability can be expressed in many forms. For example, the probability of flipping a head can be expressed as: Slide 7 / 176 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 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? Slide 8 / 176 A 0 B 1/6 C 1/2 D 1 Probability = Need Number a hint? of favorable outcomes Total Move number the box. of possible outcomes A R T H U R 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? Slide 9 / 176 A 0 B 1/6 C 1/3 D 1 Probability = Need Number a hint? of favorable outcomes Total Move number the box. of possible outcomes A R T H U R
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? Slide 10 / 176 A 0 B 1/4 C 1/10 D 10/20 Probability = Number of favorable outcomes Total Need number a hint? of possible outcomes Move the box. 4 What is the probability of rolling a 5 on a fair number cube? Slide 11 / 176 5 What is the probability of rolling an odd on a fair number cube? Slide 12 / 176
6 What is the probability of rolling a 7 on a fair number cube? Slide 13 / 176 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? Slide 14 / 176 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? Slide 15 / 176
9 Mary chooses an integer at random from 1 to 6. What is the probability that the integer she chooses is a prime number? Slide 16 / 176 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. 10 Each of the hats shown below has colored marbles placed inside. Hat A contains five green marbles and four red marbles. Hat B contains six blue marbles and five red marbles. Hat C contains five green marbles and five blue marbles. If a student were to randomly pick one marble from each of these three hats, determine from which hat the student would most likely pick a green marble. Justify your answer. Slide 17 / 176 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 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. Slide 18 / 176 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
Slide 19 / 176 Experimental & Theoretical Probability Click to go to Table of Contents Click on an object. What is the outcome? Slide 20 / 176 O Outcomt Experimental Probability Probability of an event number of times the outcome happened number of times experiment was repeated Slide 21 / 176 Answers Flip the coin 5 times and determine the experimental probability of heads. Heads Tails Pull Experimental Probability The ratio of the numbe times an event occurs total number of times t activity is performed.
Experimental Probability Slide 22 / 176 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. P(hole-in-one) = # of successes # of trials = 24 124 = 6 31 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 Slide 23 / 176 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. P(participation) = # of successes # of trials = 8 40 = 1 5 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. Sally rolled a die 10 times and the results are shown below. Slide 24 / 176 Use this information to answer the following questions. # on Die Picture of Roll Results 1 1 one 2 3 twos 3 1 three 4 0 fours 5 4 fives 6 1 six
11 What is the experimental probability of rolling a 5? Slide 25 / 176 A B C D 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 12 What is the experimental probability of rolling a 4? Slide 26 / 176 A B C D 1/2 5/4 4/4 0 # on Die Picture of Roll Results 1 1 one 2 3 twos 3 1 three 4 0 fours 5 4 fives 6 1 six 13 Based on the experimental probability you found, if you rolled the die 100 times, how many sixes would you expect to get? Picture of A # on Die 6 sixes Roll Results B C D 10 sixes 12 sixes 60 sixes 1 1 one 2 3 twos 3 4 1 three 0 fours 5 4 fives 6 1 six Slide 27 / 176 These are the results after 10 rolls of the die
Answer 14 Mike flipped a coin 15 times and it landed on tails 11 times. What is the experimental probability of landing on heads? Slide 28 / 176 heoretical robability Theoretical Probability What is the theoretical probability of spinning green? Probability Equally Lik Slide 29 / 176 Pull the tabs for definitions. Fair Theoretical Probability Slide 30 / 176 Probability of an event number of favorable outcomes total number of possible outcomes
Theoretical Probability Slide 31 / 176 Example 1 - Marbles Find the probability of randomly choosing a white marble from the marbles shown. P(white) = # of favorable outcomes # of possible outcomes = 4 2 = 10 5 There is a 2 in 5 chance of picking a white marble or a 40% possibility. Theoretical Probability Slide 32 / 176 Example 2 - Marbles Suppose you randomly choose a gray marble. Find the probability of this event. P(gray) = # of favorable outcomes # of possible outcomes = 3 10 There is a 3 in 10 chance of picking a gray marble or a 30% possibility. Theoretical Probability Slide 33 / 176 Example 3 - Coins Find the probability of getting tails when you flip a coin. P(tails) = # of favorable outcomes # of possible outcomes = 1 2 There is a 1 in 2 chance of getting tails when you flip a coin or a 50% possibility.
15 What is the theoretical probability of picking a green marble? Slide 34 / 176 A B C D 1/8 7/8 1/7 1 16 What is the theoretical probability of picking a black marble? Slide 35 / 176 A B C D 1/8 7/8 1/7 1 17 What is the theoretical probability of picking a white marble? Slide 36 / 176 A B C D 1/8 7/8 1/7 1
18 What is the theoretical probability of not picking a white marble? Slide 37 / 176 A B C D 1/8 7/8 1/7 1 19 What is the theoretical probability of rolling a three? Slide 38 / 176 A 1/2 B 3 C 1/6 D 1 20 What is the theoretical probability of rolling an odd number? Slide 39 / 176 A 1/2 B 3 C 1/6 D 5/6
21 What is the theoretical probability of rolling a number less than 5? A 2/3 Slide 40 / 176 B 4 C 1/6 D 5/6 22 What is the theoretical probability of not rolling a 2? Slide 41 / 176 A 2/3 B 2 C 1/6 D 5/6 23 Seth tossed a fair coin five times and got five heads. The probability that the next toss will be a tail is A 0 Slide 42 / 176 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.
24 Which inequality represents the probability, x, of any event happening? Slide 43 / 176 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 Class Activity Slide 44 / 176 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. Slide 45 / 176 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?
Slide 46 / 176 Word Problems Click to go to Table of Contents 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. Slide 47 / 176 Number of Outcomes in the Event Total Number of Possible Outcomes Theoretical Probability 107 1000 107/1000=0.107 0.107 100= 10.7% 317 1000 317/1000=0.317 0.317 100= 31.7% 576 1000 576/1000=0.576 0.576 100= 57.6% 107+317+576=1000 1000/1000 1000/1000=1 100= 100% Bob, the manager of the Marvelous Marble Company tells Pete that it is time to add a yellow marble to the batch. In addition, Bob tells Pete to start making the batches in equal proportion so the customer can receive an equal amount of colors in a batch. He tells Pete he needs this taken care of right away. If you were Pete, how would you use theoretical probability to solve this problem? Assume 1000 marbles per batch (red, green, blue and yellow colored marbles) Start with 1000 marbles Divide 1000 into 4 equal parts (equal colors) Each part is equal to 250 marbles Reduce to lowest terms Do you have an explanation of the probability for Bob? Click on black circle to find answer. The customer has a 1 in 4 or 25% chance of picking any color! Slide 48 / 176
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. Slide 49 / 176 What do you know about percentages to figure out the relationship of goals scored to goals attempted? 19 shots made 100 shots attempted = 19% Experimental Probability =number of times the outcome happened number of times experiment was repeated Erica's Experimental Probability = number Move of to goals Reveal number Move of to attempts Reveal Please continue on next slide... Let's estimate the number of goals Erica scored. Slide 50 / 176 Erica makes 19% of her shots on goal. About what percent would be a good estimate to use? Erica takes 1,100 shots on goal. About how many attempts did Erica take? 19 is very close to 20 100 100 so she makes about 20% of her shots on goal. 1,100 is very close to 1,000. So we will estimate that Erica has about 1,000 attempts (please click on the boxes to see if you are correct) Erica wants to find 20% of 1,000. Her math looks like this: Slide 51 / 176 Erica figures she made about 200 of her shots on goal.
Slide 52 / 176 Challenge Can you find the actual values that will give you 19%? Answer Hint Experimental Probability Example 3 - Gardening Last year, Lexi planted 12 tulip bulbs, but only 10 of them bloomed. This year she intends to plant 60 tulip bulbs. Use experimental probability to predict how many bulbs will bloom. Slide 53 / 176 10 bloom 12 total = x bloom 60 total Solve this proportion by looking at it times 5 10 bloom 12 total = 50 bloom 60 total Based on her experience last year,lexi can expect 50 out of 60 tulips to bloom. Experimental Probability Slide 54 / 176 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. 32 made 50 attempts Solve this proportion using cross products 32 75 = 50 x 2400 = 50x x made 75 attempts 48 = x Based on your performance yesterday,you can expect to make 48 free throws out of 75 attempts. =
Now, its your turn. Calculate the experimental probability for the number of goals. Slide 55 / 176 Number of attempts 100 Number of goals 30 Experimental Probability.30 or 30% 1000 500 2000 600 150 1600.60 or 60%.30 or 30%.80 or 80% 25 Tom was at bat 50 times and hit the ball 10 times. What is the experimental probability for hitting the ball? Slide 56 / 176 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. Slide 57 / 176
27 What is the theoretical probability of randomly selecting a jack from a deck of cards? Slide 58 / 176 28 Mark rolled a 3 on a die for 7 out of 20 rolls. What is the experimental probability for rolling a 3? Slide 59 / 176 29 What is the theoretical probability for rolling a 3 on a die? Slide 60 / 176
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? Slide 61 / 176 Slide 62 / 176 Fundamental Counting Principle Click to go to Table of Contents What should I wear today? Buddy has 2 shirts and 3 pairs of pants to choose from. How many different outfits can he make? Slide 63 / 176
Let's find out how many outfits Buddy can make using a tree diagram. Slide 64 / 176 Pull To make a tree diagram, match each pair of pants with each shirt. Buddy can make 6 outfits! Or we could use multiplication to find out how many outfits Buddy could make. 3 x 2 = 6 pants shirts outfits Slide 65 / 176 How many different meals can we create using the following menu? Slide 66 / 176 Side Entree Dessert Soup Lasagna Ice Cream Salad French Fries Chicken Fajita Burrito Pizza Hamburger Cake
Create a tree diagram by dragging the items. Side Entree Dessert Soup Lasagna Chicken Fajita Ice Cream Salad Burrito Pizza Cake French Fries Hamburger Slide 67 / 176 Soup Lasagna Ice Cream Cake Now try to solve the same problem using multiplication. Slide 68 / 176 Side Entree Dessert Soup Lasagna Chicken Fajita Ice Cream Salad Burrito Pizza Cake French Fries Hamburger x x = Sides Entrees Desserts Meals If you were to pick 4 digits to be your identification number, how many choices are there? Slide 69 / 176 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.
If you were to pick 4 digits to be your identification number, how many choices are there if there is no replacement? Slide 70 / 176 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. 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 23 x x x = 1 1 1 23 23 23 Total Possibilities Slide 71 / 176 4 56 4 56 4 56 4 56 7 8 9 0 7 8 9 7 8 7 5,040 combinations Move to Reveal Answer If you were to pick 4 digits to be your identification number, how many choices are there if there is replacement? Slide 72 / 176 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
Students are given a lock for their gym lockers. Each code requires you to enter 4 single digit numbers. If the numbers can be repeated, but zero cannot be the first number how many different codes are possible? x x x = 1 23 1 23 1 23 1 23 Total Possibilities Slide 73 / 176 4 56 4 56 4 56 4 56 7 8 9 7 8 9 0 7 8 9 0 7 8 9 0 9,000 combinations Move to Reveal Answer 7,893,600 combinations This cryptex has a map to treasure buried somewhere in New Jersey inside of it! Each of the 5 columns lists every letter in the alphabet once. What are the total number of codes that can be created if the letters cannot be repeated? Slide 74 / 176 Hint 1 Hint 2 Click on lock to reveal answers Hint 3 Hint 1 Hint 2 25 11,881,376 Click on lock to reveal answers This cryptex has a map to treasure buried somewhere in New Jersey inside of it! Each of the 5 columns lists every letter in the alphabet once. What is the probability of the codes containing the letters MATH (in that order) as the first 4 letters in the code? (Last letter can be a repeat) Challenge Version Hint 3 Slide 75 / 176
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 of a blouse, a skirt, and a scarf? A 8 + 6 + 5 B 8 6 5 C 8! 6! 5! Slide 76 / 176 D 19 C 3 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. 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 Slide 77 / 176 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 33 Joe has 4 different hats, 3 different shirts, and 2 pairs of pants. How many different outfits can Joe make? Slide 78 / 176 A B C D 9 outfits 14 outfits 24 outfits 12 outfits To enlarge calculator, pull the bottom right corner.
34 Stacy is trying to find out how many different combinations of license plates there. She lives in New Jersey where there are 3 letters followed by 3 numbers. How many different combinations of license plates are there? Slide 79 / 176 A B C D 17,576,000 license plates 12,812,904 license plates 729 license plates 17,576 license plates To enlarge calculator, pull the bottom right corner. 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? Slide 80 / 176 A B letter number 36 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? Slide 81 / 176 A B C D 45 choices 90 choices 21 choices 42 choices To enlarge calculator, pull the bottom right corner.
37 Ali is making bracelets for her and her friends out of beads. She figured that each bracelet should be about 10 beads. If she only has blue and green beads, how many different bracelets can she possibly make? Slide 82 / 176 A B C D 1,024 bracelets 1,000 bracelets 100 bracelets 20 bracelets To enlarge calculator, pull the bottom right corner. 38 5 styles of bikes come in 4 colors each, how many different bikes choices are available? Slide 83 / 176 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? Slide 84 / 176
40 How many ways can 3 students be named president, vice president,and secretary if each holds only 1 office? Slide 85 / 176 41 How many ways can a 4-question multiple choice quiz be answered if the there are 5 choices per question? Slide 86 / 176 42 A locker combination system uses three digits from 0 to 9. How many different three-digit combinations with no digit repeated are possible? Slide 87 / 176 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.
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? Slide 88 / 176 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. 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? Slide 89 / 176 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. 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). Slide 90 / 176 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 91 / 176 Permutations and Combinations Click to go to Table of Contents How many ways can the following animals be arranged? Slide 92 / 176 There are two methods to solve this problem: Method 1: List all the possible groupings Method 2: Use the permutation. Method 1: List all possible groupings. Slide 93 / 176 There are 24 arrangements of 4 animals in 4 positions.
Method 2: Use the permutation. A permutation is an arrangement of n objects in which order is important. Slide 94 / 176 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. 4 3 2 1 = 24 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." 46 What is the value of 5!? Slide 95 / 176 47 How many ways can the letters in FROG be arranged? Slide 96 / 176
48 In how many ways can a police officer, fireman and a first aid responder enter a room single file? Slide 97 / 176 A 3 B 3! C 6 D 6! E 1 49 In how many ways can four race cars finish a race that has no ties? Slide 98 / 176 A 4 B 4! C 24 D 24! E 12 50 How many ways can the letters the word HOUSE be arranged? Slide 99 / 176
51 How many ways can 4 books be arranged on a shelf? Slide 100 / 176 How many ways can the letters in the word DEER be rearranged? Slide 101 / 176 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 52 In how many ways can the letters in JERSEY be arranged? Slide 102 / 176
53 How many different three-letter arrangements can be formed using the letters in the word ABSOLUTE if each letter is used only once? Slide 103 / 176 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. Permutation Formula Slide 104 / 176 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 np r, where np r = 5P 3 = = = = 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. 54 Find the value of 6P 2 Slide 105 / 176
55 Find the value of 4P 1 Slide 106 / 176 56 Find the value of 5P 5 Slide 107 / 176 Hint: 0! = 1 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? Slide 108 / 176 20P 3 = = 20! = 20 19 18 17! 17! 17! = 6840
Find the number of permutations of 4 objects taken 3 at a time. Slide 109 / 176 How many 4-digit numbers can you make using each of the digits 1, 2, 3, and 4 exactly once? 4 P 4 = 4! = 4 3 2 1 = 24 0! 1 57 10 cars are in a race. How many ways can prizes be awarded for first, second and third place? Slide 110 / 176 58 How many ways can four out of seven books be arranged on a shelf? Slide 111 / 176
59 You are taking 7 classes, three before lunch. How many possible arrangements are there for morning classes? Slide 112 / 176 60 The teacher is going to select a president and vicepresident from the 24 students in class. How many possible arrangements are there for president and vice-president? Slide 113 / 176 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 order the toppings were placed. Can you think of other examples when order does not matter?
61 You must read 5 of the 10 books on the summer reading list. This is an example of a Slide 115 / 176 A B Combination Permutation 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 B Combination Permutation Slide 116 / 176 63 10 people are in a room. How many different pairs can be made? This is an example of a Slide 117 / 176 A B Combination Permutation
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 Slide 118 / 176 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 119 / 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! Slide 120 / 176 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
66 Find the number of combinations. Slide 121 / 176 5 C 2 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? Slide 122 / 176 68 There are 75 flowers in the shop. How many different arrangements containing 10 flowers can be created? Slide 123 / 176
69 Eight people enter the chess tournament. How many different pairings are possible? Slide 124 / 176 70 Mary can select 3 of 5 shirts to pack for the trip. How many different groupings are possible? Slide 125 / 176 71 How many different three-member teams can be selected from a group of seven students? A 1 B 35 C 210 D 5040 Slide 126 / 176
Slide 127 / 176 Probability of Compound Events Click to go to Table of Contents Probability of Compound Events Slide 128 / 176 First - decide if the two events are independent or dependent. When the outcome of one event does not affect the outcome of another event, the two events are independent. Use formula: Probability (A and B) = Probability (A) Probability (B) Independent Example Slide 129 / 176 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
When the outcome of one event affects the outcome of another event, the two events are dependent. Slide 130 / 176 Use formula: Probability (A & B) = Probability(A) Probability(B given A) Dependent Example Select a card from a deck of cards, do not replace it in the deck, shuffle the deck, and select a second card. What is the probability that you will pick a 6 and then a king? P (6 and a king) = P(6) P(king given a six has been selected) 4 4 = 4 52 51 663 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? Slide 131 / 176 73 A lottery machine generates numbers randomly. Two numbers between 1 and 9 are generated. What is the probability that both numbers are 5? Slide 132 / 176
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? Slide 133 / 176 75 What is the probability that the first two cards drawn from a full deck are both hearts? (without replacement) Slide 134 / 176 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? Slide 135 / 176
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? Slide 136 / 176 78 At a school fair, the spinner represented in the accompanying diagram is spun twice. Slide 137 / 176 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? A C B 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. 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? Slide 138 / 176 A C B 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.
80 The probability that it will snow on Sunday is. Slide 139 / 176 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 C B 2 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 Click to go to Table of Contents Events are mutually exclusive or disjoint if they have no outcomes in common. Slide 141 / 176 Example: Event A: Roll a 3 Event B: Roll an even number Event A 3 Event B 2 4 6
Slide 142 / 176 Overlapping Events are events that have one or more outcomes in common Example Event A: Roll an even number Event B: Roll a number greater than 3 Event A 2 4 6 Event B 5 81 Are the events mutually exclusive? Event A: Selecting an Ace Event B: Selecting a red card Yes No Slide 143 / 176 82 Are the events mutually exclusive? Event A: Rolling a prime number Event B: Rolling an even number Yes No Slide 144 / 176
83 Are the events mutually exclusive? Event A: Rolling a number less than 4 Event B: Rolling an even number Yes No Slide 145 / 176 84 Are the events mutually exclusive? Event A: Selecting a piece of fruit Event B: Selecting an apple Yes No Slide 146 / 176 85 Are the events mutually exclusive? Event A: Roll a multiple of 3 Event B: Roll a divisor of 19 Yes No Slide 147 / 176
86 Are the events mutually exclusive? Event A: Roll a multiple of 3 Event B: Roll a divisor of 19 Yes No Slide 148 / 176 87 Are the events mutually exclusive? Randomly select a football card Event A: Select a Philadelphia Eagle Event B: Select a starting quarterback Yes Slide 149 / 176 No 88 Are the events mutually exclusive? Event A: The Yankees won the World Series Event B: The Mets won the National League Pennant Slide 150 / 176 Yes No
Take notes! Formula probability of two mutually exclusive events P(A or B) = P(A) + P(B) Slide 151 / 176 Pull What is the probability of drawing a 5 or an Ace from a standard deck of cards? Slide 152 / 176 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... Check your answer by pulling down the screen. P(5 or A) = P(5) + P(A) 4 + 4 = 8 52 52 52 reduce 2 13 Slide 153 / 176 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)
A bag contains the following candy bars: 3 Snickers 4 Mounds 2 Almond Joy 1 Reese's Peanut Butter Cup Slide 154 / 176 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 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 or Pepsi? Slide 155 / 176 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? Slide 156 / 176
91 A die is rolled twice. What is the probability that a 4 or an odd number is rolled? Slide 157 / 176 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? Slide 158 / 176 93 Events A and B are disjoint. Find P(A or B). P(A) = P(B) = Slide 159 / 176
94 Events A and B are disjoint. Find P(A or B). P(A) = P(B) = Slide 160 / 176 What's the problem with this situation... Slide 161 / 176 Think about this... Find is the probability of selecting a black card or a 7? P(black or 7) If the situation is 2 events CAN occur at the same time, then these are NOT mutually exclusive events. Slide 162 / 176 Take notes! Formula probability of two events which are NOT mutually exclusive P(A or B) = P(A) + P(B) - P(A and B)
What is the probability of selecting a black card or a 7? P (black or 7) Slide 163 / 176 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 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 Slide 164 / 176 girls lacrosse Pull Now do the math! 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. Slide 165 / 176 P(girl or lacrosse) = P(girl) + P(lacrosse) - P(girl and lacrosse) 121 300 + 29 300-25 300 125 300 = 0.416
95 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? Slide 166 / 176 A P(odd) + P(multiple of 3) B P(odd) x P (multiple of 3) - P(odd and multiple of 3) C P(odd) x P(multiple of 3) D P(odd) + P (multiple of 3) - P(odd and multiple of 3) 96 Events A and B are overlapping. Find P(A or B). P(A) = P(B) = P(A and B) = Slide 167 / 176 97 Events A and B are overlapping. Find P(A or B). P(A) = P(B) = P(A and B) = Slide 168 / 176
98 What is the probability of rolling a number less than two or an odd number? Slide 169 / 176 99 What is the probability of rolling a number that is not even or that is not a multiple of 3? Slide 170 / 176 Slide 171 / 176 Complementary Events Click to go to Table of Contents
Complementary Events Slide 172 / 176 Two events are complementary events if they are mutually exclusive and one event or the other must occur. The sum of the probabilities of complementary events is always 1. P(A) + P(not A) = 1 Example: The forecast calls for a 30% chance of rain. What is the probability that it will not rain? P(rain) + P(not rain) = 1.3 +? = 1 P(not rain) =.7 100 Given P(A), find P(not A). P(A) = 52% P(not A) = % Slide 173 / 176 101 Given P(A), find P(not A). P(A) = Slide 174 / 176 P(not A) =
102 The spinner below is divided into eight equal regions and is spun once. What is the probability of not getting red? Slide 175 / 176 Green Yellow Red Red A B Blue White Red Purple 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. 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 C Slide 176 / 176 B 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.