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•
If you want to have the best chances of getting a red gumball from a gumball machine, is it better if the
machine is full of gumballs or half empty? How do the chances of getting an ace in a deck of playing
cards change if you have three or four decks of cards to choose from instead of only one deck? In this
lesson, you will think about the size of the sample space (the collection of all possible outcomes of an
event). Think about these questions as you work today:
•
•
How has the “portion” or part we are interested in changed?
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How has the “whole” or total changed?
Has the event become more or less likely?
1-75. Your team will be given a bag containing a set of colored blocks or counters.
Alternatively, use the 1-75 Student eTool (CPM) which contains a bag with 1 yellow, 2 red, 4 green, and
5 blue blocks. Each team will receive a bag that is identical to yours.
a. Look at the blocks in your bag. If you were to reach into the bag and select one block without
looking, what is the likelihood that it would be:
i.
Red?
ii.
Green?
iii.
Blue?
iv.
Orange?
b. Do your answers for part (a) represent theoretical or experimental probabilities? Justify your
response.
•
1-76. If you were to select one block from the bag 12 times, replacing the block you drew between each
selection, how many of those times would you expect to have selected a blue block? What if you drew
24 times? Discuss both situations with your team and explain your answers.
•
1-77. DOUBLING BAGS
•
Now imagine that you and another team have combined the blocks from both of your bags into one bag.
Explore this concept using the 1-77 Student eTool (CPM) where two bags from 1-75 are combined.
.
Do you think the larger sample space will change the likelihood of drawing blocks of different
colors? Discuss this with your team and be ready to explain your ideas to the class.
a. Get a second bag of blocks from your teacher and combine the contents of both bags. How
many total blocks are there in the bag now? How many are there of each color?
b. Work with your team to find the theoretical probability for selecting each color of block in the
combined bags.
c. Has the probability for drawing each different-colored block changed? How do your answers for
part (c) above compare to the theoretical probabilities that you calculated for the original bag in
problem 1-75? With your team, make sense of how the probability for drawing a blue block
compares before and after combining the bags.
d. If you were to make 12 draws from the combined bag, replacing the block between draws, how
many times would you expect to draw a blue block? Explainwhy your answer makes sense. .
•
1-78. In problems 1-75 through 1-77, even though you combined bags or changed the number of
selections you made, the probability of drawing a blue block remained the same.
.
Do you think the probabilities would change if you combined three bags? Why or why not?
a. What change do you think you could make in order to increase the chances of choosing a blue
block? Explain your reasoning.
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1-79. Additional Challenge: Use the spinner from the Lesson 1.2.2 Resource Page or the 1-79 Student
eTool (CPM). Play a game with the spinner while keeping score as follows:
•
Every time you spin purple, you lose two points.
•
Every time you spin green, you get one point.
•
Every time you spin orange, you get three points.
If you play this game for a long time, do you think it is more likely that you will end up with a positive
score or a negative score? Make a prediction and then try it out. You may want to keep score with
counters.
Probability Vocabulary and Definitions
•
Outcome: Any possible or actual result of the action considered, such as rolling a 5 on a standard
number cube or getting tails when flipping a coin.
•
Event: A desired (or successful) outcome or group of outcomes from an experiment, such as rolling an
even number on a standard number cube.
•
Sample Space: All possible outcomes of a situation. For example, the sample space for flipping a coin
is heads and tails; rolling a standard number cube has six possible outcomes (1, 2, 3, 4, 5, and 6).
•
Probability: The likelihood that an event will occur. Probabilities may be written as fractions,
decimals, or percents. An event that is guaranteed to happen has a probability of 1, or 100%. An event
that has no chance of happening has a probability of 0, or 0%. Events that “might happen” have
probabilities between 0 and 1 or between 0% and 100%. In general, the more likely an event is to
happen, the greater its probability.
•
Experimental Probability: The probability based on data collected in experiments.
•
•
Experimental Probability =
Theoretical Probability is a calculated probability based on the possible outcomes when they all have
the same chance of occurring.
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Theoretical Probability =
.
In the context of probability, “successful” usually means a desired or specified outcome (event), such as
rolling a 2 on a number cube (probability of ). To calculate the probability of rolling a 2, first figure
out how many possible outcomes there are. Since there are six faces on the number cube, the number of
possible outcomes is 6. Of the six faces, only one of the faces has a 2 on it. Thus, to find the probability
of rolling a 2, you would write:
•
•
or
or approximately 16.7%.
1-80. Tom keeps all of his favorite marbles in a special leather bag. Right now, five red marbles, four
blue marbles, and three yellow marbles are in the bag. Homework Help ✎
a. If he randomly chooses one marble to give to a friend, what is the probability that it is blue?
b. Tom does not really want to give away blue marbles and would like to change the probability
that he chooses a blue marble to
. How many marbles that are not blue could he add to the
bag so that the probability of choosing a blue marble becomes
•
?
1-81. Your team is in charge of games at the CPM Amusement Park. One of the
games involves a robotic arm that randomly grabs a stuffed animal out of a large bin. You need to set
up the game so that the probability of a customer’s grabbing a teddy bear is exactly
Help ✎
.
. Homework
How would you set up the bin? Explain.
a. What if you returned to check on the bin and found that there were 4 teddy bears left and 12
other animals? What could you add to or remove from the bin to return the probability of
selecting a teddy bear to
?
•
1-82. Write four different fractions that are equal to 1. Use your calculator to check that you are
correct. Homework Help ✎
•
1-83. A rectangular park is 150 yards on one side and 125 yards on the other. Homework Help ✎
.
If Debbie walks around the park two times, how far does she walk? Sketch a figure and show
your work.
a. If Debbie wanted to walk 1,000,000 yards, how many times would she have to walk around the
park?
•
1-84. Find the perimeter and area of each figure. Homework Help ✎
.
a.