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© 2011 THE CARNEGIE FOUNDATION FOR THE ADVANCEMENT OF TEACHING A PATHWAY THROUGH STATISTICS, VERSION 1.5, STATWAY™ - STUDENT HANDOUT

STATWAY™ STUDENT HANDOUT

Lesson 6.1.2 Probability Rules

STUDENT NAME DATE

INTRODUCTION Buttoning Up Probability

Now that you have a basic understanding of probability as the chance of success in the long run, you can

work on more complex problems. To help you visualize the outcomes, you will use bags of buttons with

different characteristics. Keep in mind that the buttons could represent similar classifications; for example, a

shipment of new file cabinets that have two drawers or four drawers and have dents and scratches or no

dents and scratches.

TRY THESE – PART 1 1 First, organize the data in the table. Carefully remove buttons from the bag and separate them

into the listed categories.

White Black Red Total

Two holes

Four holes

Total

A Is it possible to make two separate groups of buttons, one containing all the buttons with

four holes and the other with all the red buttons? If no, why not? B Is it possible to make two separate groups of buttons, one containing all the red buttons and

the other with all the white buttons? If no, why not?

STATWAY STUDENT HANDOUT | 2



C If you can make two separate groups, the two characteristics are mutually exclusive. Which of

the following are mutually exclusive? Two holes and four holes Black and four holes

2 Sort the buttons as directed or use the table to perform the required calculation.

A Separate out the red buttons. Find the probability that a randomly selected button is red. B Separate out the four-holed buttons. Find the probability that a randomly selected button

has four holes. C Separate out the buttons that are red or have four holes. Find the probability that a randomly

selected button is red or has four holes. Note that some buttons have both characteristics, but they should only be counted once.

D Find the probability that a randomly selected button is red and has four holes. E Briefly explain why the probabilities in Questions 2c and 2d are different. F Find the probability that a randomly selected button is white or has 2 holes. G Find the probability that a randomly selected button is black or red.




TRY THESE – PART 2 3 Sort the buttons as directed or use the table to perform the required calculation.

A Separate out the white buttons. Find the probability that a randomly selected button is white.

B Separate out the buttons with four holes. Find the probability that a randomly selected

button is white, given that it has four holes. Or you could think of it this way: using the four-holed buttons as the total, how many the four-holed buttons are white.

YOU NEED TO KNOW

This is called a conditional probability. You want to find the probability that a button is white. However,

there is an extra condition—the button must have four holes. So all other buttons—the ones with two

holes—are irrelevant. You only choose from the group with four holes. That is the group you are given to

choose from. The number of four-holed buttons becomes the new total and is the denominator of the

fraction. The probability can be written as P(white given four holes)or P(white|four holes) to simplify your

writing.

C Compare the two probabilities in Questions 3a and 3b. D Separate out the buttons with four holes. Find the probability that a randomly selected

button has four holes. E Find the probability that a button has four holes given it is white.




F Compare the two probabilities in Questions 3d and 3e.

YOU NEED TO KNOW When the probability of an event is the same when you are given another characteristic, the two

characteristics or events are said to be independent. So knowing that a button is white does not change the

probability that it has four holes, we can say that the button being white is independent of it having two

holes.

4 If we wanted to find out if a button having two holes and being black are independent events,

what would we need to know? A Find the two probabilities you identified in (4). B Draw your conclusion. Are the events of the button having two holes and being black

independent?







Questions 5 and 6 will help you understand why independence is such a big deal.

5 Sort the buttons as directed or use the table to perform the required calculation. A Separate out all the buttons that are both white and have four holes. Find the probability

that a randomly selected button comes from this group when drawn from the entire group.

B Multiply the probability a button is white by the probability a button has four holes. Note: You have already found these probabilities in Question 3.

C Compare your answers in Questions 5a and 5b. D Are the events (characteristics) of the button being white and having two holes independent?

6 Sort the buttons as directed or use the table to perform the required calculation.

A Separate out all the buttons that are both black and have two holes. Find the probability that

a randomly selected button comes from this group when drawn from the entire group. B Multiply the probability that a button is black with the probability that a button has two

holes. C Compare the two probabilities in Questions 6a and 6b.




D Are the events (characteristics) of the button being black and having two holes independent?

YOU NEED TO KNOW Independence

If two events are independent, the probability of both events occurring is equal to the product of the

individual probabilities.

Example: What is the probability of rolling a fair die two times and getting two sixes? Since each roll is

independent, the probabilities remain the same.

P(2 sixes) =

Example: What is the probability that an adult American male is at least 6 feet tall and has an intelligence

quotient (IQ) of at least 120, given that the probability a man is at least 6 feet tall is 0.145 and the

probability of an adult having an IQ of at least 120 is 0.089?

Since height and IQ are independent events, you can simply multiply the two individual probabilities:

P(at least 6 feet tall) = 0.145

P(IQ at least 120) = 0.089

P(at least 6 feet tall and IQ at least 120) = 0.145 × 0.089 = 0.0129

If events are not independent, you cannot simply multiply the individual probabilities. You must consider

the additional information given and how that affects the probability of the additional events.

Example: What is the probability a three-person subcommittee contains all females if the members are

randomly selected from a group with five males and five females?

Since the probability of choosing a female changes with each additional selection, the events are not

independent. For the selection of the first person, there are 10 people in the group: five females and five

males, so the probability of selecting a female is

P(female first selection)= 5

10




For the selection of the second person, there are nine people left in the group: four females and five males,

so the probability of selecting a female is

P(female secondselection)= 4

9

For the selection of the third person, there are eight people left in the group: three females and five males,

so the probability of selecting a female is

P(female thirdselection)= 3

8

So, the probability of selecting an all-female committee is equal to the product of the individual probabilities

above:

P(three females)= 5

10́

4

9́

3

8= 0.08333

TRY THESE – PART 3 With and Without Replacement 7 Sort the buttons as directed or use the table and perform the required calculation.

A Place all the buttons in the bag and find the probability that you get a randomly selected

button that is red on two consecutive draws, if you replace the first button before drawing the second.

B Are the events of selecting two red buttons independent if we return the first button to the

bag prior to selecting the second button?




C With all the buttons in the bag, consider randomly drawing two buttons that are red without replacing the first draw.

D Are the events of selecting two red buttons independent if we are not returning the first

button prior to selecting the second button? E Find the probability of drawing three red buttons with and without replacement. Note that

replacement means after the third draw you are still holding only one red button. Without replacement, you have a group of three red buttons together. Whenever you are choosing a group, it is understood there is no replacement.

TRY THESE – PART 4 Complements 8 Find the probabilities below.

A P(red) = B P(not red) = C P(red) + P(not red)=

YOU NEED TO KNOW




Since the two probabilities make up the entire set of buttons, you can find either probability by subtracting

the other from 1. Red and not red are complements of each other.

NEXT STEPS – PART 5 9 If the chance of rain is 40%, what is the chance of no rain? Explain your answer.

10 Write a statement which is the complement of “students who earned a grade of A on their exam”.




TAKE IT HOME

1 A salesman for energy-efficient radiant barrier attic insulation believes a homeowner with a newer

roof is more likely to purchase his product. Looking over past records of sales agents soliciting

businesses by ringing doorbells and discussing their products, he sorts the sales records into three

groups: customers with a 30-year roof in good condition, those with a 20-year roof in good

condition, and those in need of a new roof. His results are in the following table.

30-year Roof 20-year Roof

In Need of a New Roof

Total

Purchased insulation 8 5 4 17

Did not make a purchase 42 77 94 213

Total 50 82 98 230

Does knowing the condition of a prospective customer’s roof help the salesman determine who is

a more likely candidate for an insulation sale? What would you advise a door-to-door salesman of

the radiant barrier insulation? Why? Justify your answer using appropriate probabilities and the

concept of independence.




2 Adult rattlesnakes tend to hide rather than bite humans. Out of pure curiosity humans often go in

search of rattlesnakes. Bite records seem to show that rattlesnakes are aware that humans are

too big to eat. Medical records of rattlesnake bites show that about 30% of the bites contain no

venom. These are known as dry bites. Note: Consider rattlesnake bites independent events. A What is the probability that two rattlesnake victims get dry bites?

B If you are bitten by a rattlesnake, what is the probability the bite contains venom? C If an emergency center in West Texas sees six patients with rattlesnake bites during the fall

rattlesnake round-up, what is the probability the first five bites contain venom and the last one do not contain venom?

+++++ This lesson is part of STATWAY™, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. Version 1.0, A Pathway Through Statistics, Statway™ was created by the Charles A. Dana Center at the University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version 1.5 and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is an open-resource research and development community that seeks to harvest the wisdom of its diverse participants in systematic and disciplined inquiries to improve developmental mathematics instruction. For more information on the Statway Networked Improvement Community, please visit carnegiefoundation.org. For the most recent version of instructional materials, visit Statway.org/kernel.

+++++ STATWAY™ and the Carnegie Foundation logo are trademarks of the Carnegie Foundation for the Advancement of Teaching. A Pathway Through College Statistics may be used as provided in the CC BY

http://www.carnegiefoundation.org/




license, but neither the Statway trademark nor the Carnegie Foundation logo may be used without the prior written consent of the Carnegie Foundation.

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Lesson 6.1.2 Probability Rules - tlok.org 6/Students/PDF/lesson_6.1.2_version_1.5...Lesson 6.1.2 Probability Rules STUDENT NAME DATE INTRODUCTION Buttoning Up Probability Now that