Introduction to Probability Click to go to Table of Contents - - PDF document
Slide 1 / 188 Slide 2 / 188 7th Grade Statistics and Probability 2015-11-19 www.njctl.org Slide 3 / 188 Table of Contents Click on a topic to go to that section. Introduction to Probability Experimental and Theoretical Sampling
2015-11-19 www.njctl.org
· Introduction to Probability · Experimental and Theoretical · Word Problems · Probability of Compound Events
Click on a topic to go to that section.
· Sampling · Measures of Center · Measures of Variation · Mean Absolute Deviation · Glossary
· Introduction to Probability · Experimental and Theoretical · Word Problems · Probability of Compound Events
Click on a topic to go to that section.
· Sampling · Measures of Center · Measures of Variation · Mean Absolute Deviation · Glossary
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Teacher Notes
Vocabulary Words are bolded in the presentation. The text box the word is in is then linked to the page at the end
word defined on it.
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One way to express probability is to use a fraction.
P(event)
Example: What is the probability of flipping a nickel and the nickel landing on heads? Step 1: What are the possible outcomes? Step 2: What is the number of favorable outcomes? Step 3: Put it all together to answer the question. The probability of flipping a nickel and landing on heads is: 1 2
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clickExample: What is the probability of flipping a nickel and the nickel landing on heads? Step 1: What are the possible outcomes? Step 2: What is the number of favorable outcomes? Step 3: Put it all together to answer the question. The probability of flipping a nickel and landing on heads is: 1 2
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Answer
The outcomes become your denominator. P(heads) = The favorable
your numerator. P(heads) =
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 0.5 2 The probability of randomly selecting a blue marble can be expressed as: 1 or 1:6 or 16.7% or .167 6
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 i b l e U n l i k e l y Equally Likely L i k e l y 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).
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
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
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
Probability = Number of favorable outcomes Total number of possible outcomes
Click for hint
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
Probability = Number of favorable outcomes Total number of possible outcomes
Click for hint
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Answer
B
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
Probability = Number of favorable outcomes Total number of possible outcomes
Click for hint
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
Probability = Number of favorable outcomes Total number of possible outcomes
Click for hint
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Answer
C
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 0 B 1/4 C 1/10 D 10/20 Probability = Number of favorable outcomes Total number of possible outcomes
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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 0 B 1/4 C 1/10 D 10/20 Probability = Number of favorable outcomes Total number of possible outcomes
Click for hint
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Answer
B
4 What is the probability of rolling a 5 on a fair number cube?
4 What is the probability of rolling a 5 on a fair number cube?
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Answer
1/6
5 What is the probability of rolling a composite number on a fair number cube?
5 What is the probability of rolling a composite number on a fair number cube?
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Answer
1/3
6 What is the probability of rolling a 7 on a fair number cube?
6 What is the probability of rolling a 7 on a fair number cube?
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Answer
7 You have black, blue, and white t-shirts in your
1/3 and the probability of picking a blue t-shirt is 1/2, what is the probability of picking a white t-shirt?
7 You have black, blue, and white t-shirts in your
1/3 and the probability of picking a blue t-shirt is 1/2, what is the probability of picking a white t-shirt?
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Answer
1/6
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?
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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Answer
1/25
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
June, 2011.
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
June, 2011. [This object is a pull tab]
Answer
B
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
A Hat B Hat C Hat
From the New York State Education Department. Office of Assessment Policy, Development and
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
A Hat B Hat C Hat
From the New York State Education Department. Office of Assessment Policy, Development and
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Answer
Hat A P(green) = 5/9 Hat B P(green) = 0 Hat C P(green) = 1/2
Determine the fewest number of marbles, if any, and the color
probability of picking a green marble will be one-half in each
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.
From the New York State Education Department. Office of Assessment Policy, Development and
June, 2011
Determine the fewest number of marbles, if any, and the color
probability of picking a green marble will be one-half in each
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.
From the New York State Education Department. Office of Assessment Policy, Development and
June, 2011 [This object is a pull tab]
Answer
Hat A needs 1 red marble. Hat B needs 11 green marbles. Hat C does not need marbles.
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Click on an object. What is the outcome?
number of times the outcome happened
number of times experiment was repeated
Flip the coin 5 times and determine the experimental probability of heads. Probability
Heads Tails
Answer
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.
P(hole-in-one) = # of successes # of trials = 24 124 = 6 31
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.
P(participation) = # of successes # of trials = 8 40 = 1 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
Sally rolled a die 10 times and the results are shown below. Use this information to answer the following questions.
11 What is the experimental probability of rolling a 5? A 1/2 B 5/4 C 4/5 D 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 Answer
12 What is the experimental probability of rolling a 4? A 1/2 B 5/4 C D 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 Answer
13 Based on the experimental probability you found, if you rolled the die 100 times, how many sixes would you expect to get? A 6 sizes B 10 sixes C 12 sixes D 60 sixes
These are the results after 10 rolls of the die Answer
# on Die Picture of Roll Results 1 1 one 2 3 twos 3 1 three 4 0 fours 5 4 fives 6 1 six
14 Mike flipped a coin 15 times and it landed on tails 11
14 Mike flipped a coin 15 times and it landed on tails 11
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Answer
4/15
What is the theoretical probability of spinning green? Is this a fair probability?
What is the theoretical probability of spinning green? Is this a fair probability?
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Answer
1/4
Probability
number of favorable outcomes
total number of possible outcomes
Example 1 - Marbles Find the theoretical 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 = =
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 =
There is a 1 in 2 chance of getting tails when you flip a coin or a 50% possibility.
Example 3 - Coins Find the probability of getting tails when you flip a coin. P(tails) = # of favorable outcomes # of possible outcomes 1 2 =
15 What is the theoretical probability of picking a green marble? A 1/8 B 7/8 C 1/7 D 1 R R G W W Y Y B
15 What is the theoretical probability of picking a green marble? A 1/8 B 7/8 C 1/7 D 1 R R G W W Y Y B
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Answer
A
16 What is the theoretical probability of picking a black marble? A 1/8 B 7/8 C 1/7 D R R G W W Y Y B
16 What is the theoretical probability of picking a black marble? A 1/8 B 7/8 C 1/7 D R R G W W Y Y B
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Answer
D
17 What is the theoretical probability of picking a white marble? A 1/8 B 7/8 C 1/4 D 1 R R G W W Y Y B
17 What is the theoretical probability of picking a white marble? A 1/8 B 7/8 C 1/4 D 1 R R G W W Y Y B
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Answer
C
18 What is the theoretical probability of not picking a white marble? A 3/4 B 7/8 C 1/7 D 1 R R G W W Y Y B
18 What is the theoretical probability of not picking a white marble? A 3/4 B 7/8 C 1/7 D 1 R R G W W Y Y B
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Answer
A
19 What is the theoretical probability of rolling a three? A 1/2 B 3 C 1/6 D 1
19 What is the theoretical probability of rolling a three? A 1/2 B 3 C 1/6 D 1
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Answer
C
20 What is the theoretical probability of rolling an odd number? A 1/2 B 3 C 1/6 D 5/6
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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Answer
A
21 What is the theoretical probability of rolling a number less than 5? A 2/3 B 4 C 1/6 D 5/6
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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Answer
A
22 What is the theoretical probability of not rolling a 2? A 2/3 B 2 C 1/6 D 5/6
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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Answer
D
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
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
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Answer
D The result of the next toss does not depend
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
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
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Answer
D
25 The spinner shown is divided into 8 equal sections. The arrow on this spinner is spun once. What is the probability that the arrow will land on a section labeled with a number greater than 3? Enter only your fraction.
From PARCC EOY sample test calculator #1
25 The spinner shown is divided into 8 equal sections. The arrow on this spinner is spun once. What is the probability that the arrow will land on a section labeled with a number greater than 3? Enter only your fraction.
From PARCC EOY sample test calculator #1
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Answer
26 Reagan will use a random number generator 1,200
statement best predicts how many times the digit 5 will appear among the 1,200 results? A It will appear exactly 200 times. B It will appear close to 200 times but probably not exactly 200 times. C It will appear exactly 240 times. D It will appear close to 240 times but probably not exactly 240 times.
From PARCC EOY sample test calculator #17
26 Reagan will use a random number generator 1,200
statement best predicts how many times the digit 5 will appear among the 1,200 results? A It will appear exactly 200 times. B It will appear close to 200 times but probably not exactly 200 times. C It will appear exactly 240 times. D It will appear close to 240 times but probably not exactly 240 times.
From PARCC EOY sample test calculator #17
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Answer
B
· 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.
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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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.
A sample is considered 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?
random (or unbiased) when every possible sample of the same size has an equal chance of being
may not be reliable.
A sample is considered 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?
random (or unbiased) when every possible sample of the same size has an equal chance of being
may not be reliable.
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Answer
includes people who take the train and does not include people who may walk, drive, or bike.
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.
Determine whether the situation would produce a random sample. You want to find out about music preferences of people living in your area. You and your friends survey every tenth person who enters the mall nearest you.
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Answer
Yes, this situation would produce a random sample.
27 Food services at your school wants to increase the number of students who eat hot lunch in the cafeteria. They conduct a survey by asking the first 20 students that enter the cafeteria lunch line to determine the students' preferences for hot lunch. Is this survey reliable? Explain your answer. Yes No
27 Food services at your school wants to increase the number of students who eat hot lunch in the cafeteria. They conduct a survey by asking the first 20 students that enter the cafeteria lunch line to determine the students' preferences for hot lunch. Is this survey reliable? Explain your answer. Yes No
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Answer
entering the cafeteria lunch line most likely already buy a hot lunch.
28 The guidance counselors want to organize a career
end in a 7 about their grades and career counseling
sample? Explain your answer. Yes No
28 The guidance counselors want to organize a career
end in a 7 about their grades and career counseling
sample? Explain your answer. Yes No
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Answer
Yes
29 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
29 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
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Answer
No
How would you estimate the size of a crowd? What methods would you use? Could you use the same methods to estimate the number
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
Suppose this represents all the wolves on the mountain. One way to estimate the number of wolves on a mountain is to use the capture-recapture method.
Wildlife biologists first find some wolves and tag them.
Then they release them back onto the mountain.
They wait until all the wolves have mixed together.Then they find a second group of wolves and count how many are tagged.
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
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
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?
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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Answer
= 315 x 600 1,500 787.5 = x about 788 people
30 Eight hundred sixty 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?
30 Eight hundred sixty out of 4,000 people surveyed watched Dancing with the Stars. How many people in the US watched if there are 93.1 million people?
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Answer
860 x 4,000 93,100,000 80,066,000,000 = 4,000x 20,016,500 = x 20,016,500 people watched =
31 Six out of 150 tires need to be realigned. How many out of 12,000 are going to need to be realigned?
31 Six out of 150 tires need to be realigned. How many out of 12,000 are going to need to be realigned?
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Answer
6 x 150 12,000 72,000 = 150x 480 = x 480 tires =
32 You are an inspector. You find 3 faulty bulbs out of 50. Estimate the number of faulty bulbs in a lot of 2,000.
32 You are an inspector. You find 3 faulty bulbs out of 50. Estimate the number of faulty bulbs in a lot of 2,000.
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Answer
120 faulty bulbs
33 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?
33 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?
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Answer
about 542 votes
34 The chart shows the number of people wearing different types of shoes in Mr. Thomas' English
high-top sneakers. Explain your reasoning.
Number of Students Low-top sneakers 12 High-top sneakers 7 Sandals 3 Boots 6
Shoes
Answer
35 Josephine owns a diner that is open every day for breakfast, lunch, and dinner. She offers a regular menu and a menu with specials for each of the three meals. She wanted to estimate the percentage of her customers that order form the menu with specials. She selected a random sample of 50 customers who had lunch at her diner during a three-month period. She determined that 28% of these people ordered for the menu with specials. Which statement about Josephine's sample is true? A The sample is the percentage of customers who order from the menu with specials. B The sample might not be representative of the popultation because it only included lunch customers. C The sample shows that exactly 28% of Josephine's customers order from the menu with specials. D No generalizations can be made from this sample, because the sample size of 50 is too small.
From PARCC EOY sample test calculator #13
35 Josephine owns a diner that is open every day for breakfast, lunch, and dinner. She offers a regular menu and a menu with specials for each of the three meals. She wanted to estimate the percentage of her customers that order form the menu with specials. She selected a random sample of 50 customers who had lunch at her diner during a three-month period. She determined that 28% of these people ordered for the menu with specials. Which statement about Josephine's sample is true? A The sample is the percentage of customers who order from the menu with specials. B The sample might not be representative of the popultation because it only included lunch customers. C The sample shows that exactly 28% of Josephine's customers order from the menu with specials. D No generalizations can be made from this sample, because the sample size of 50 is too small.
From PARCC EOY sample test calculator #13
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Answer
B
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.
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
Answer
The student council would like to sell potato chips at the next basketball game to raise money. They surveyed some students to figure out how many packages of each type of potato chip they would need to buy. For home games, the expected attendance is approximately 250 spectators. Use the chart to answer the next three questions.
Student Sample Regular BBQ Cheddar #1 8 10 7 #2 8 11 6
36 How many students participated in each survey?
Student Sample Regular BBQ Cheddar #1 8 10 7 #2 8 11 6 Answer
37 According to the two random samples, which flavor potato chip should the student council purchase the most of? A Regular B BBQ C Cheddar
Student Sample Regular BBQ Cheddar #1 8 10 7 #2 8 11 6 Answer
38 Use the first random sample to evaluate the number of packages of cheddar potato chips the student council should purchase.
Student Sample Regular BBQ Cheddar #1 8 10 7 #2 8 11 6 Answer
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19 shots made 100 shots attempted = 19% Example 1 - Soccer 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 number of goals number of attempts Erica's Experimental Probability = Move to Reveal Move to Reveal
click to reveal
19 100 20 100 is very close to so she makes about 20%
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
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
What are the actual values that will give you 19%?
Challenge
Remember sometimes it helps to turn a percent into a decimal prior to solving the problem.
Click for hint
What are the actual values that will give you 19%?
Challenge
Remember sometimes it helps to turn a percent into a decimal prior to solving the problem.
Click for hint
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Answer
0.19 1,100 = 210 Erica made 210 of her 1,100 shots. 0.19 19% =
Example 2 - Gardening Last year, Lexi planted 12 tulip bulbs, but only 10
Use experimental probability to predict how many bulbs will bloom. Based on her experience last year,Lexi can expect 50
Solve this proportion by equivalent fractions.
10 bloom 12 total x bloom 60 total = 10 bloom 12 total 50 bloom 60 total =
Example 3 - 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.
32 75 = 50 x 2400 = 50x 48 = x 32 made 50 attempts x made 75 attempts =
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.
39 Tom was at bat 50 times and hit the ball 10 times. What is the experimental probability for hitting the ball?
39 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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Answer 1 or 0.2 or 20% 5
40 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.
40 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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Answer
50
41 What is the theoretical probability of randomly selecting a jack from a deck of cards?
41 What is the theoretical probability of randomly selecting a jack from a deck of cards?
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Answer
1/13
42 Mark rolled a 3 on a die for 7 out of 20 rolls. What is the experimental probability for rolling a 3?
42 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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Answer 7 or 0.35 or 35% 20
43 Mark rolled a 3 on a die for 7 out of 20 rolls. What is the theoretical probability for rolling a 3?
43 Mark rolled a 3 on a die for 7 out of 20 rolls. What is the theoretical probability for rolling a 3?
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Answer 1 or 0.16 or 16.7% 6
44 Some books are laid on a desk. Two are English, three are mathematics, one is French, and four are social
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?
44 Some books are laid on a desk. Two are English, three are mathematics, one is French, and four are social
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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Answer
1/8
45 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
45 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
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Answer
C
46 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
46 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
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Answer
A
Lindsey would like to know the number of people at a movie theater that will buy a movie ticket and popcorn. Based on past data, the probability that a person who is selected at random from those that buy movie tickets and also buy popcorn is 0.6. Lindsey designs a simulation to estimate the probability that exactly two in a group of three people selected randomly at a movie theater will buy both a movie ticket and popcorn. For the simulation Lindsey used a number generator that generates random numbers. · Any number from 1 through 6 represents a person who buys a movie ticket and popcorn. · Any number from 7 through 9
buys only a movie ticket. · Use info for next two questions.
From PARCC EOY sample test calculator #3
47 Part A In the simulation, one result was "100". What does this result simulate? A No one in a group of three randomly-chosen people who buy movie tickets also buys popcorn. B Exactly one person in a group of three randomly-chosen people who buy movie tickets also buys popcorn. C Exactly two people in a group of three randomly-chosen people who buy movie tickets also buy popcorn. D All three people in a group of three randomly-chosen people who buy movie tickets also buy popcorn.
47 Part A In the simulation, one result was "100". What does this result simulate? A No one in a group of three randomly-chosen people who buy movie tickets also buys popcorn. B Exactly one person in a group of three randomly-chosen people who buy movie tickets also buys popcorn. C Exactly two people in a group of three randomly-chosen people who buy movie tickets also buy popcorn. D All three people in a group of three randomly-chosen people who buy movie tickets also buy popcorn.
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Answer
Part A: B
48 Part B Use the results of the simulation to estimate the probability that exactly two of three people selected at random from those who buy movie tickets will also buy popcorn.
48 Part B Use the results of the simulation to estimate the probability that exactly two of three people selected at random from those who buy movie tickets will also buy popcorn.
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Answer
Part B
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For the probability of compound events, first - decide if the two events are independent or dependent. When the outcome of one event does not affect the outcome of another event, the two events are independent. Use formula: Probability (A and B) = Probability (A) Probability (B)
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. 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?
P (6 and a king) = P(6) P(king given a six has been selected) 4 4 = 4 52 51 663 Notice your demominator when down by 1. Why?
Independent Dependent Try to name some other independent and dependent events.
49 The names of 6 boys and 10 girls from your class are put in a hat. What is the probability that the first two names chosen will both be boys? (w/o replacement)
49 The names of 6 boys and 10 girls from your class are put in a hat. What is the probability that the first two names chosen will both be boys? (w/o replacement)
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Answer 6 x 5 = 1 16 15 8
50 A lottery machine generates numbers randomly. Two numbers between 1 and 9 are generated. What is the probability that both numbers are 5?
50 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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Answer 1 x 1 = 1 9 9 81
51 The TV repair person is in a room with 20 broken TVs. Two sets have broken wires and 5 sets have a faulty computer chip. What is the probability that the first TV repaired has both problems?
51 The TV repair person is in a room with 20 broken TVs. Two sets have broken wires and 5 sets have a faulty computer chip. What is the probability that the first TV repaired has both problems?
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Answer 2 x 5 = 1 20 20 40
52 What is the probability that the first two cards drawn from a full deck are both hearts? (without replacement)
52 What is the probability that the first two cards drawn from a full deck are both hearts? (without replacement)
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Answer 13 x 12 = 1 52 51 17
53 A spinner containing 5 colors: red, blue, yellow, white and green is spun and a die, numbered 1 through 6, is
rolling a two?
53 A spinner containing 5 colors: red, blue, yellow, white and green is spun and a die, numbered 1 through 6, is
rolling a two?
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Answer 1 x 1 = 1 5 6 30
54 A drawer contains 5 brown socks, 6 black socks, and 9 navy blue socks. The power is out. What is the probability that Sam chooses two socks that are both black?
54 A drawer contains 5 brown socks, 6 black socks, and 9 navy blue socks. The power is out. What is the probability that Sam chooses two socks that are both black?
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Answer 6 x 5 = 3 20 19 38
56 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
56 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
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Answer A 5 x 2 = 10 7 6 42
57 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
57 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
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Answer C 3 ÷ 3 = 1 10 5 2
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Sometimes we can make general statements about a set of data as shown in this first question.
58 Alexis chose a random sample of 10 jars of almonds from each of two different brands, X and Y. Each jar in the sample was the same
are shown in the plots.
A The number of almonds in jars from Brand X tends to be greater and more consistent than those from Brand Y. B The number of almonds in jars from Brand X tends to be greater and less consistent than those from Brand Y. C The number of almonds in jars from Brand X tends to be fewer and more consistent than those from Brand Y. D The number of almonds in jars from Brand X tends to be fewer and less consistent than those from Brand Y.
From PARCC EOY sample test calculator #7
58 Alexis chose a random sample of 10 jars of almonds from each of two different brands, X and Y. Each jar in the sample was the same
are shown in the plots.
A The number of almonds in jars from Brand X tends to be greater and more consistent than those from Brand Y. B The number of almonds in jars from Brand X tends to be greater and less consistent than those from Brand Y. C The number of almonds in jars from Brand X tends to be fewer and more consistent than those from Brand Y. D The number of almonds in jars from Brand X tends to be fewer and less consistent than those from Brand Y.
From PARCC EOY sample test calculator #7
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Answer
B
Other times we will make statements about the data based on measure of center and variation that we can calculate. This will be the topics for the rest of this chapter.
Median - The middle data value when the values are written in numerical order Mean (Average) - The sum of the data values divided by the number of items Mode - The data value that occurs the most often
Joey wanted to convince his mom to give him some money for a snack from the concession stand. Below are the prices of the different snacks. $1.75, $0.75, $1.25, $0.75, $2.50, $2.00
What is the mean of this data set? $1.75, $0.75, $1.25, $0.75, $2.50, $2.00 Step 1: Add up all of the numbers. 1.75 + 0.75 + 1.25 + 0.75 + 2.50 + 2.00 = 9.00 Step 2: Divide the sum by the number of items listed. 9.00 / 6 = 1.50 The mean cost of concession stand snacks is $1.50.
Find the mini mean 1.25 + 1.75 = 1.50 2 What do you do when you have two numbers left? (click)
What is the median of this data set? $1.75, $0.75, $1.25, $0.75, $2.50, $2.00 Step 1: Order the numbers from least to greatest. 0.75, 0.75, 1.25, 1.75, 2.00, 2.50 Step 2: Find the middle value. 0.75, 0.75, 1.25, 1.75, 2.00, 2.50 The median cost of concession stand snacks is $1.50.
What is the mode of this data set? $1.75, $0.75, $1.25, $0.75, $2.50, $2.00 Step 1: Look for the number that appears most often. 1.75, 0.75, 1.25, 0.75, 2.50, 2.00 The mode cost of concession stand snacks is $0.75.
Mean
$1.50
Median
$1.50
Mode
$0.75
How can Joey use this information to ask his mom for money? $1.75, $0.75, $1.25, $0.75, $2.50, $2.00
Time Spent Texting Daily by 7th Grade Students (in minutes)
60 15 30 45 75 90 105 120
Girls
60 15 30 45 75 90 105 120
Boys
Use the dot plots to compare the 2 samples.
60 15 30 45 75 90 105 120
Girls
Find the mean, median, and mode for the sample of girls.
60 15 30 45 75 90 105 120
Girls
Find the mean, median, and mode for the sample of girls.
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Answer
Mean 88.5 min or 1 hr 28 min and 30 sec *Remember to find out how many seconds 0.5 is equal to multiply the decimal by how many seconds are in a
Median 90 minutes or 1 hour 30 minutes Mode 60 minutes or 1 hour
60 15 30 45 75 90 105 120
Boys
Find the mean, median, and mode for the sample of boys.
60 15 30 45 75 90 105 120
Boys
Find the mean, median, and mode for the sample of boys.
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Answer Mean 40.5 minutes or 40 minutes and 30 seconds Median 30 seconds Mode 30 minutes and 60 minutes
Girls Boys Mean 88.5 40.5 Median 90 30 Mode 60 30 and 60
Now compare the two measures of center. Make a statement about the average time spent texting daily by 7th grade students. Answer
59 What is the mean of the stem-and-leaf plot? Stem Leaf 1 1 1 2 2 0 0 3 5 5 4 8 Key: 1 | 1 = 11
59 What is the mean of the stem-and-leaf plot? Stem Leaf 1 1 1 2 2 0 0 3 5 5 4 8 Key: 1 | 1 = 11
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Answer
25
60 What is the median of the stem-and-leaf plot? Stem Leaf 1 1 1 2 2 0 0 3 5 5 4 8 Key: 1 | 1 = 11
60 What is the median of the stem-and-leaf plot? Stem Leaf 1 1 1 2 2 0 0 3 5 5 4 8 Key: 1 | 1 = 11
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Answer
20
61 What is the mode of the stem-and-leaf plot? Stem Leaf 1 1 1 2 2 0 0 3 5 5 4 8 Key: 1 | 1 = 11
61 What is the mode of the stem-and-leaf plot? Stem Leaf 1 1 1 2 2 0 0 3 5 5 4 8 Key: 1 | 1 = 11
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Answer
11 and 20
62 What is the mean of the stem-and-leaf plot? Stem Leaf 1 8 9 3 7 7 9 Key: 1 | 8 = 1.8
62 What is the mean of the stem-and-leaf plot? Stem Leaf 1 8 9 3 7 7 9 Key: 1 | 8 = 1.8
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Answer
3
63 What is the median of the stem-and-leaf plot? Stem Leaf 1 8 9 3 7 7 9 Key: 1 | 8 = 1.8
63 What is the median of the stem-and-leaf plot? Stem Leaf 1 8 9 3 7 7 9 Key: 1 | 8 = 1.8
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Answer
3.7
64 What is the mode of the stem-and-leaf plot? Stem Leaf 1 8 9 3 7 7 9 Key: 1 | 8 = 1.8
64 What is the mode of the stem-and-leaf plot? Stem Leaf 1 8 9 3 7 7 9 Key: 1 | 8 = 1.8
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Answer
3.7
Miss M's Math Class Scores
90 50 60 70 80 100
1st Period Scores
Use the dot plots to find the measures of center.
Miss M's Math Class Scores
90 50 60 70 80 100
1st Period Scores
Use the dot plots to find the measures of center.
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Answer
Mean: about 80.417 Median: 80 Mode: 80
Miss M's Math Class Scores
90 50 60 70 80 100
8th Period Scores
Use the dot plots to find the measures of center.
Miss M's Math Class Scores
90 50 60 70 80 100
8th Period Scores
Use the dot plots to find the measures of center.
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Answer
Mean: about 75.83 Median: 75 Mode: 75
Write a statement comparing the averages of Miss M's 1st period class scores to her 8th period class scores.
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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) Range - The difference between the greatest data value and the least data value.
1, 5, 8, 3, 2, 5, 2, 8, 9, 5 1, 2, 2, 3, 5, 5, 5, 8, 8, 9 · To find the interquartile range of the data set, we first have to find the quartiles. Step 1 : Order the numbers from least to greatest.
1, 2, 2, 3, 5, 5, 5, 8, 8, 9 Step 2 : Find the median. 5 median *Note : · If the median falls in between two data values, all of the values are still used to calculate the upper and lower quartiles. · If the median falls exactly on one of the two data values, than that values is NOT used to calculate the upper and lower quartiles.
1, 2, 2, 3, 5, 5, 5, 8, 8, 9
5 median 8 Upper 2 Lower Quartile Quartile
Step 3 : Find the upper and lower quartiles. Find the mean of each half of the data set.
Upper Lower Quartile Quartile Step 4 : Subtract the lower quartile from the upper quartile. Interquartile = Range
= 6
1, 2, 2, 3, 5, 5, 5, 8, 8, 9 · To find the range, subtract the least value from the greatest value. Greatest Least Value Value = Range
= 8
1, 2, 2, 3, 5, 5, 5, 8, 8, 9
5 Median 8 Upper 2 Lower Quartile Quartile 1 Least Value 9 Greatest Value
These 5 values are used to create a box-and-whisker
and then connect them to look like a box with whiskers on both sides.
click just above the number line to reveal
1 2 3 4 5 6 7 8 9 10
65 What is the median of the data set?
1 2 3 4 5 6 7 8 9 10
65 What is the median of the data set?
1 2 3 4 5 6 7 8 9 10
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Answer
4
66 What is the interquartile range using the given information? Least Value = 3 Lower Quartile = 6 Median = 7 Upper Quartile = 10 Greatest Value = 11
66 What is the interquartile range using the given information? Least Value = 3 Lower Quartile = 6 Median = 7 Upper Quartile = 10 Greatest Value = 11
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Answer
4
67 What is the range for the following data set? 3, 5, 10, 4, 2, 2, 1
67 What is the range for the following data set? 3, 5, 10, 4, 2, 2, 1
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Answer
9
68 What is the interquartile range for the following data set? 3, 5, 10, 4, 2, 2, 1
68 What is the interquartile range for the following data set? 3, 5, 10, 4, 2, 2, 1
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Answer
3
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Mean absolute deviation
data value and the mean.
Find the mean absolute deviation of the following data. Quiz Scores 65, 75, 90, 90, 100 Step 1: Find the mean. 65 + 75 + 90 + 90 + 100 = 420 = 84 5 5
Step 2: Find the absolute deviation. To do this you need to subtract the mean and each data point. Then take the absolute value of each difference.
Step 3: Find the mean absolute deviation (MAD). To do this find the mean using the absolute deviation numbers. 19 + 9 + 6 + 6 + 16 5 = 56 = 11.2 5 The MAD is 11.2 points. Answer
Girls' Team Boys' Team
The number of goals scored by the players on the boys' and girls' LAX teams are displayed below. Compare the variability of the mean goals scored for both teams.
Girls' Team
5 6 4 3 2 1 7 8 9 10Boys' Team
Step 1: Find the mean for each team.
Girls' Team
5 6 4 3 2 1 7 8 9 10Boys' Team
Step 1: Find the mean for each team.
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Answer
= 48 = 6 8 2 + 5 + 6 + 6 + 6 + 7 + 8 + 8 8 Boys' Team Mean = 40 = 5 8 3 + 3 + 4 + 5 + 5 + 5 + 5 + 10 8 Girls' Team Mean
Goals Mean Deviation Absolute Mean Dev. Goals Mean Deviation Absolute Mean Dev.
Girls' Team Boys' Team
Comparing Two Data Sets
Step 2: Find the absolute deviations.
Girls' Team Boys' Team
Step 3: Find the mean absolute deviations.
Girls' Team Boys' Team
Step 3: Find the mean absolute deviations.
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Answer
= 10 = 1.25 8 2 + 2 + 1 + 0 + 0 + 0 + 0 + 5 8 Girls' Team MAD = 10 = 1.25 8 4 + 1 + 0 + 0 + 0 + 1 + 2 + 2 8 Boys' Team MAD
Comparison Statements 1.25 = 1.25 The variability is equal for both the boys and girls LAX teams. On average, the boy players scored 1 more goal than the girl players. (How do you know this?)
Comparison Statements 1.25 = 1.25 The variability is equal for both the boys and girls LAX teams. On average, the boy players scored 1 more goal than the girl players. (How do you know this?)
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Answer
Remember when comparing averages, look at the means. Girls = 5 goals Boys = 6 goals
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 Use the following data to answer the next seven questions.
69 What is the mean number of pages per chapter in the Hunger Games?
69 What is the mean number of pages per chapter in the Hunger Games?
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Answer
13.63 pages
70 What is the mean number of pages per chapter in Twilight?
70 What is the mean number of pages per chapter in Twilight?
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Answer
20.41 pages
71 What is the difference of the means?
71 What is the difference of the means?
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Answer
6.78
72 What is the mean absolute deviation of the data set for Hunger Games? (Hint: Round mean to the nearest ones.)
72 What is the mean absolute deviation of the data set for Hunger Games? (Hint: Round mean to the nearest ones.)
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Answer
1.78
73 What is the mean absolute deviation of the data set for Twilight? (Hint: Round mean to the nearest ones.)
73 What is the mean absolute deviation of the data set for Twilight? (Hint: Round mean to the nearest ones.)
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Answer
4.32
74 Which book has more variability in the number of pages per chapter? A Hunger Games B Twilight
74 Which book has more variability in the number of pages per chapter? A Hunger Games B Twilight
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Answer
B
75 On average, there are ______ pages per chapter in the Hunger Games than in Twilight. A more B less
75 On average, there are ______ pages per chapter in the Hunger Games than in Twilight. A more B less
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Answer
B less
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Teacher Notes
Vocabulary Words are bolded in the presentation. The text box the word is in is then linked to the page at the end
word defined on it.
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Asking only flight attendants if they believe flying is safe. Asking everyone in Hershey Park if they like chocolate. Asking everyone at ComicCon if they like comic books.
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A method of sampling that is used to try and estimate the entire
released into the wild. Later a second sample of animals are caught to compute using a ratio the amount of tagged animals to the population as a whole.
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The prob. of flipping heads AND rolling 4 on a die. The prob. of selecting a Jack OR a 3 card. The prob. of selecting a Jack AND a 3 card.
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Probability (A & B) = Prob(A) *Prob(B given A) The prob. of selecting a Jack AND a 3 card.
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Sides on a Coin
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number of times the outcome happened number of times experiment was repeated
Probability
Last week, 24 out
hole-in-one on the last hole. Find the experimental probability that a golfer makes this shot.
P(hole-in-one) = # of successes = # of trials 24 124 = 6 31
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Tossing a Coin Rolling a Fair Die Spinning a Fair Spinner
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Probability (A and B) = Prob(A)Prob(B) The prob. of flipping heads AND rolling 4 on a die.
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25% 25%
25% 25%
Q1 Q2 Q3
1,3,3,4,5,6,6,7,8,8
Q1 Q2 Q3
1 2 3 4 5 6 7 8
Q1
Q2 Q3
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25% 25%
25% 25%
Q1 Q2 Q3
1,3,3,4,5,6,6,7,8,8
Q1 Q2 Q3
1 2 3 4 5 6 7 8
Q1
Q2 Q3
Median
Median
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Set of Data:
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Find the mean Subtract the mean from each data point Find the mean of the differences
3-2=1 4-3=1 3-3=0
3-2=1 4-3=1
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Median is 2.5
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The mode is 4.
The mode is 4 and 2.
There is no mode.
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Population Sample
· All m&ms in a bag · All types of dogs in a dog park · All students wearing glasses in a classroom
NOT just people in a place
Back to Instruction What is the probability
the nickel landing on heads?
1 favorable 2 possible
Number of favorable outcomes Total number of possible outcomes
Probability
P(event)
1 or 50% 2 1:2 or 0.5 Many Forms!
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25% 25%
25% 25%
Q1 Q2
Q3 1,3,3,4,5,6,6,7,8,8
Q1 Q2
Q3
1 2 3 4 5 6 7 8
Q1
Q2 Q3
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Asking everyone in a classroom if they believe flying is safe. Asking everyone in a classroom if they like chocolate. Asking everyone in a classroom if they like comic books.
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The range is 10.
The range is 35.
The range is 17.
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Population Sample
·
in a bag ·
park ·
glasses in a classroom
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number of favorable outcomes total number of possible outcomes
Probability
Find the probability of getting tails when you flip a coin.
P(tails) =
# of favorable
# of possible
1 2 =
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25% 25%
25% 25%
Q1 Q2
Q3 1,3,3,4,5,6,6,7,8,8
Q1 Q2 Q3
1 2 3 4 5 6 7 8
Q1
Q2 Q3
Median
Median