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
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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?
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
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.
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.
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.
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