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COMPOUND PROBABILITY
INTRODUCTION
The objective for this lesson on Compound Probability is, the student will find probabilities of
compound events using organized lists, tables, tree diagrams, and simulations.
The skills students should have in order to help them in this lesson include, Simple Probability
and Relative Frequency.
We will have three essential questions that will be guiding our lesson. Number one, what is
compound probability? Number two, how does compound probability differ from simple
probability? Number three, explain how compound probability can be calculated.
Begin by completing the warm-up on Simple Probability to prepare for this lesson on Compound
Probability.
SOLVE INTRODUCTION
Leslie has a fair number cube labeled one through six, and she is trying to figure out what the
chances are of rolling specific numbers on the cube. What is the probability that Leslie will roll
a two or a three on the number cube?
S, Study the Problem. Underline the question. This problem is asking me to find the probability
that Leslie will roll a two or a three on the number cube.
O, Organize the Facts. First we identify the facts within the problem. Then we eliminate any
unnecessary facts. And last we list the necessary facts. Six sides labeled one through six. A
favorable outcome is rolling a two or a three.
L, Line up a Plan. Write in words what your plan of action will be. Identify the probability of
rolling a two or three with the number cube.
Choose an operation or operations. Use fractions to create probabilities.
V, Verify Your Plan with Action. Estimate your answer. Less than one but more than zero.
Carry out your plan. The probability or rolling a two or three: Two out of the six possible
outcomes are favorable. The probability of rolling a two or three is: two over six which is
equivalent to one over three.
E, Examine Your Results.
Does your answer make sense? Compare your answer to the question. Yes, because I found the
probability Leslie will roll a two or a three.
Is your answer reasonable? Compare your answer to the estimate. Yes, because my answer is
between zero and one.
Is your answer accurate? Check your work. Yes.
Write your answer in a complete sentence. The probability or rolling a two or a three is two over
six or one over three.
EXTEND THE SOLVE PROBLEM – COMPOUND PROBABILITY WITH A LIST
In the SOLVE problem on the previous page, we determined that the probability of rolling a two
or three on the number cube was one third. Part of Leslie’s math project was to work with the
probability of two separate events and determine the probability of certain outcomes. The
teacher asked Leslie to add flipping a red and yellow counter as a second event with the desired
outcome of a red.
How many events are involved in this scenario? Two
What are the two events? The probability of rolling a two or three, and then the probability of
flipping a red.
Explain how we determined the probability of rolling a two or three. We created a fraction with
a numerator of the number of favorable outcomes over the number of total possible outcomes.
How can we determine the probability of flipping the counter and having it land on red? Create
a fraction with the number of favorable outcomes over the total number of possible outcomes.
What is the probability of flipping a counter and having it land on red? One out of two, or one
half.
What is another word for all possible outcomes? Sample space
Working in pairs, create a list of the possible outcomes of the first event and then the second
event. How many total outcomes are possible? Twelve
How many outcomes have a two or three and Red? Circle and list them. Two – two red and
three red.
What is similar about finding the probability using a list and using the fraction method? You
still need to know the total number of outcomes and the favorable outcomes. Any other
similarities?
How did we find the probability? We placed the number of favorable outcomes over the total
possible outcomes.
What is the probability of rolling a two or three and then the counter landing on red? Two out of
twelve which is equivalent to one over six.
Remember to simplify your answers!
COMPOUND PROBABILITY WITH A TREE DIAGRAM
Mia is rolling a number cube and pulling marbles out of a bag. The bag contains one red marble,
one green marble, and one blue marble. If Mia rolls the number cube and then picks one marble
out of the bag, what is the probability that she will roll a number greater than four and pick a
blue marble?
What do you notice about the problem at the top of the page? It is the same problem from the
previous page.
We can also determine the probability of multiple events using a second method.
Identify the possible outcomes for the toss of the number cube. One, two, three, four, five and
six
Below each of the possible outcomes for the first event, list the three possibilities for the second
event.
The letter R represents red, G is for green and B is for blue.
Where can you look on the tree diagram to determine the number of possible outcomes? At the
bottom row.
How many outcomes are there? Eighteen
Which numbers are greater than four? Five and six
How many of the branches represent a possible outcome of a number greater than four and a blue
marble? Circle them. Two favorable outcomes
How do we determine the probability? Place the number of favorable outcomes over the total
possible outcomes.
What is the probability of rolling a number greater than four and choosing a blue marble? Two
out of eighteen, which is equivalent to one out of nine.
COMPARING SIMPLE AND COMPOUND PROBABILITY
Let’s review the three events in the graphic organizer.
These are all of the events that we used in the previous problems we completed. The SOLVE
Problem, and the event was rolling a two or three. Leslie’s Problem, and the events were rolling
a two or three and Flipping a Red. And the last event is Mia’s Problem where she was trying to
roll a number greater than four and pick a blue marble.
In Column One, how many events were there in the SOLVE problem? One
What type of probability do we have when there is only one event? Simple
In Columns two and three, how many events were there to consider? Two
When we have more than one event what do we call this? Compound Probability
What method did we use to determine the probability in Column one? Create a fraction of
favorable outcomes over total outcomes.
What method did we use to determine the probability in Column two? Create a list of possible
outcomes; create a fraction of favorable outcomes over total outcomes.
What method did we use to determine the probability in Column three? Create a list of possible
outcomes and a tree diagram; create a fraction of favorable outcomes over total outcomes.
Record the methods in the graphic organizer.
What is similar between the simple probability scenario and the compound probability scenario?
In both cases, we still found the favorable outcomes over the total outcomes.
What are some differences between scenarios with simple and compound probability?
Compound probabilities involve more than one event, and we found it helpful to use lists and
tree diagrams to help us find all of the favorable outcomes as well as all of the possible
outcomes.
Now, let’s complete the next SOLVE problem.
SOLVE PROBLEM – COMPOUND PROBABILITY
Lashelle is playing a game where she must spin a spinner and roll a die. The spinner contains
three equal sections colored blue, red and yellow. The die has eight sides labeled one through
eight. What is the probability that when Lashelle spins she will not land on blue and will roll a
number five or greater?
S, Study the Problem. Underline the question.
This problem is asking me to find the probability of not spinning blue and rolling a number five
or greater.
O, Organize the Facts. First identify each fact in the word problem. Then eliminate any
unnecessary facts. And list the necessary facts. There are three equal sections: blue, red and
yellow and an eight-sided die labeled one through eight.
L, Line Up a Plan. Write in words what your plan of action will be. Create a tree diagram
showing all of the outcomes of the two events and then find the favorable outcomes and the
possible outcomes.
Choose an operation or operations. Create a fraction with favorable outcomes over total
outcomes.
V, Verify Your Plan with Action. First estimate your answer. More than zero, but less than onehalf.
Then carry out your plan. We have the tree diagram for each of the possible outcomes and we’re
going to circle red and yellow because we wanted to choose not blue. Then we’re looking for
any value that’s five or greater. We circle all those and we find out there are eight favorable
outcomes that are not blue, but are five or greater.
There are twenty-four total outcomes so our answer is eight over twenty-four or one over three.
E, Examine Your Results
Does your answer make sense? Compare your answer to the question. Yes, because I found the
probability of not spinning blue and rolling a number five or greater.
Is your answer reasonable? Compare your answer to the estimate. Yes, because I estimated that
the probability would be less than one-half but more than zero.
Is your answer accurate? Check your work. Yes
Write your answer in a complete sentence. The probability of not spinning blue and rolling a
number five or greater is one-third.
EXTEND THE SOLVE PROBLEM – MULTIPLICATION WITH COMPOUND
PROBABILITY
What is the first event for Lashelle’s game scenario? Not spinning blue on the spinner.
What is the number of favorable outcomes for the first event? Two
What are the favorable outcomes? Spinning yellow or red
What is the total number of possible outcomes for the first event? Three
What is the second event? Rolling a number five or greater on the eight-sided die.
What is the number of favorable outcomes for the second event? Four
What are the favorable outcomes? Five, six, seven, or eight
What is the total number of possible outcomes for the second event? Eight
Fill that information in your chart.
Take a look at the SOLVE problem we just completed.
What was the probability of not spinning blue and rolling a number of five or greater? The
probability of not spinning blue and rolling five or greater was eight out of twenty four, or onethird.
What do you notice about the relationship between the numerators of the individual events and
the numerator of the compound probability? Multiplying the probability of the two numerators
of the single events has a product that is equal to the numerator of the compound probability
before simplifying.
What do you notice about the relationship between the denominators of the individual events and
the denominator of the compound probability? Multiplying the probability of the two
denominators of the single events has a product that is equal to the denominator of the compound
probability before simplifying.
What conclusion can we draw about the probabilities of the individual events and the probability
of the compound event? Multiplying the probability of each individual event together will give
us the probability of the compound event.
A bag of marbles contains four blue, three green, five red, and two yellow marbles. You want to
find the probability of choosing a blue marble and then choosing a red marble without replacing
the first marble chosen.
How is this situation similar to the situation on the previous page? We have two events,
choosing a red marble and choosing a blue marble.
How is this situation different from the situation on the previous page? This situation involves
only one bag of marbles, and the second choice will be made without replacing the first marble.
How will this affect the denominator of the second event? The denominator will be one less
because the marble will not be replaced.
EXPLORING COMPOUND PROBABILITY
At this time, each student pair should have a plastic cup containing an assortment of red, yellow
and blue centimeter cubes.
Using the actual cubes that are in your cup, identify the probability of choosing each
combination of cube colors. The cup contains one blue, three yellow, and two red cubes. In the
table, the vertical column of colors will represent the first cube chosen and the horizontal row of
colors will represent the second cube chosen. For all probabilities, assume that the cube will not
be replaced after chosen.
How can you find the probability of each of the compound events that will occur? Multiply the
probability of choosing the first color by the probability of choosing the second color. The
denominator for the second probability will be one less for all of the events because we assume
the first marble isn’t replaced. Also, we must decrease the numerator by one if it is a compound
event where we are choosing the same color twice.
Without completing any calculations, what will be the probability of choosing two blues in a
row? The probability will be zero. Explain how you know this. There is only one blue cube in
the cup, so without replacement, there is no chance of picking a second blue cube from the cup.
Let’s take a look at the probability of each of the compound events in the chart.
Which color combinations do you expect to actually occur most often based on your table? The
largest fractions appear with yellow then yellow, yellow then red, and red then yellow.
Complete ten trials where you choose two cubes, without replacing the first one, and record the
number of times where each combination of colors is chosen using tally marks. Place tally
marks under the column labeled “Tallies.”
What did we create by recording all of our data in the table? A frequency table
When we conduct an experiment where we are actually able to carry out the experiment this is
called a simulation.
How can we calculate the probability of each combination occurring? Record the probabilities
for each compound event under the “Relative Frequency” column. Favorable over Total: A
fraction can be created for each combination by placing the total number of tally marks for that
combination over the total number of trials, which was ten.
Now that we have found the probability and the relative frequencies of the compound events
occurring, let’s write them in the chart above so that we can compare. To make comparing
easier, let’s change all of the probability values to decimals.
How do we change the fractional probabilities to decimals? Divide the numerator of each
fraction by the denominator and record the decimal.
Fill in the table by writing the decimal form of both the probability and the relative frequency.
How do your probabilities compare with the relative frequencies of the events? Does anyone
have actual outcomes that are close to the probability? Far from the probability?
How did the actual outcome of choosing red and then yellow compare to the actual outcome of
your neighbors?
How did your outcome of choosing two blue compare with the probability? How did it compare
with your neighbors’ results? All probabilities were zero. It was impossible for more than one
blue to occur without replacement, so every group had the same result for this particular
compound event.
SOLVE PROBLEM – COMPOUND PROBABILITY
David has a drawer with several socks that are mixed and not paired. There are six black socks,
eleven white socks, seven blue socks, and nine brown socks. If David randomly chooses a sock,
does not replace it and then choose another one, what is the probability that he will choose two
white socks?
S, Study the Problem.
Underline the question.
This problem is asking me to find the probability of choosing two white socks without replacing
the first sock.
O, Organize the Facts.
First identify the facts.
Then eliminate any unnecessary facts.
List the necessary facts. Six black, eleven white, seven blue, nine brown. Choosing two socks
without replacing the first sock. In other words choosing white and then white.
L, Line Up a Plan.
Write in words what your plan of action will be. Create a fraction for each of the individual
events and multiply then together to find the overall probability.
Choose an operation or operations. Multiplication
V, Verify Your Plan with Action
Estimate your answer. More than zero but less than one
Carry out your plan.
The probability of picking the first white sock is eleven out of thirty three or one third.
The probability of picking a second white sock is ten out of thirty two or five over sixteen
The probability of choosing two white socks without replacement is five over forty-eight.
E, Examine Your Results
Does your answer make sense? Compare your answer to the question. Yes, because I found the
probability of choosing two white socks without replacement.
Is your answer reasonable? Compare your answer to the estimate. Yes, because I estimated that
the probability would be less than one but more than zero.
Is your answer accurate? Check your work. Yes.
Write your answer in a complete sentence. The probability of choosing two white socks without
replacement is five over forty-eight.
SOLVE CLOSURE – PROBABILITY FOLDABLE
Label the fourth flap “Compound Probability.” On the inside complete the section for
Compound Probability with the given information.
CLOSURE
Now let’s go back and discuss the essential questions from this lesson.
Our first question was, what is compound probability? Compound probability involves more
than one event.
Number two, how does compound probability differ from simple probability? Compound
probability involves more than one event while simple probability only involves one event.
Calculating compound probability involves two separate events while simple probability only
involves one.
And number three, explain how compound probability can be calculated. Compound probability
is calculated by multiplying the probability of the individual events together.