Download Title: Proportions

Level H Lesson 30
Simple and Compound Probability
In Lesson 30 the objective is the student will investigate simple and compound
probability.
We will have four essential questions that will be guiding our lesson. Number one:
What is simple probability? Number two: What is the difference between theoretical
and experimental probability? Number three: What is compound probability? Number
four: What is the difference between independent and dependent events?
The solve problem for our lesson is: Evie is conducting a probability experiment in her
math class. She is rolling a fair number cube and spinning a spinner which is divided
into eight equal sections numbered 1 through 8. What is the probability that Evie will
roll an even number on the number cube and spin a number greater than 5 on the
spinner?
At the beginning of the lesson we’re going to complete the S step, study of the
problem. We’re going to begin by underlining the question. What is the probability
that Evie will roll an even number on the number cube and spin a number greater
than 5 on the spinner? We are then going to complete the statement: This problem is
asking me to find the probability that Evie will roll an even number on the number
cube and spin a number greater than 5 on the spinner?
Sample space is the set of all the possible outcomes of an activity or experiment. In
order to predict the probability of an event, students need to determine all possible
outcomes. We’re going to be using some centimeter cubes and we’re going to start
by placing one cube in the first column. We’re then going to place a green cube and a
white cube in the second column. The box with the red cube on the left represents
the first event. The question is how many ways could the even occur? Since there’s
only one cube there, there is only one way the event could occur, that would be
choosing the red. The green and white cubes represent the second event. Because
we have 2 cubes to choose from: the question is how many ways could this even
occur? And the answer is 2. We’re now going to look in row 2. I’m going to be starting
with a red and a blue cube. In my second column I’m going to place a green and a
white cube. The red and the blue represent our first event. How many ways could the
event occur? Because there are two cubes to choose from, there are two ways the
events could occur. In our second column we have a green and a white cube again.
Because there are two different cubes, there’s two ways the events could occur.
Let’s go back to problem one in the first row. We know with our red cube, there’s one
way that event could occur and with our green and white cubes there’s two ways that
event could occur. In column three we’re going to draw a pictorial representation of
the ways the events could occur. I’m going to start by drawing an R to represent my
red cube. Our red cube was our first event. What can be combined with our red cube
from the first event by choosing something from the second event? The first thing we
could combine with the red would be a green. Is that the only possible outcome of
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our events, that you could have red and green? No, we could also have red and then
white. For problem two, let’s look at the pictorial representation of choosing our
cubes. We’re going to start by choosing and R, or a red cube. What in our second
column can be combined with the red cube? Well, we could combine it with a green
cube. Is that the only possible outcome of red? No, if we chose red we could also
choose white as our second outcome. Blue is also a possible choice in our first event.
If we chose blue in our first event, one of the possible outcomes of our second event
would be green. Is that the only possible outcome in the second event? No, we could
also choose blue and then choose white.
Let’s go back to problem 1 and let’s look at the fourth column. I’m going to start by
writing the word red. I’m going to make a list of the possible outcomes. We could
have red and then we could have green. We could also have red and then we could
have white. What is the number of possible outcomes in our events? We could have
red, green or we could have red, white. That means there are two possible outcomes
for our events.
Let’s look at problem 2. We’re going to start by writing the word red. Our possible
outcome with red was red and green or red and white. When we started with blue we
could have an outcome of blue and green or blue and white. So, we have four
possible outcomes. A sample space has just been recorded to show there are four
possible outcomes.
We’re now going to be working with sample space, tree diagrams and lists. We’re
going to have two different events. We have choosing shorts, either blue or tan, and
choosing a shirt: white, black, red, or green. How many different ways could our first
event occur? The shorts are the first event. We could have two different possibilities.
We could have either blue shorts or tan shorts. We’re going to call those two items of
our first event our trunk of our tree diagram. The second event is going to be
choosing the shirt. How many different ways could the second event occur? 1,2,3,4.
There are 4 different ways that second event could occur. The shirts are going to be
the branches of our tree diagram. I’ve written the four colors of shirts that possible
next to the blue shorts and also next to the tan shorts. We extend a line from the
blue shorts to each of the colors to represent those branches of our trunk. Again,
with the tan shorts, there’s a line drawn to each of the colors of the shirts. In the
second column we’re going to use our tree diagram to make a list. Blue shorts, white
shirt. Blue shorts, black shirt. Blue shorts, red shirt. Blue shorts, green shirt. We use
our second pair of shorts. Tan shorts, white shirt. Tan shorts, black shirt. Tan shorts,
red shirt. Tan shorts, green shirt. In the last column we’re going to talk about how we
can use the information from our tree diagram and list to use a numerical way to
determine the possible number of outcomes. How many different colors shorts are
listed? We have 2 different colors of shorts. How many different shirts are there?
There are 4 different colored shirts. We know that from our list and from our tree
diagram that that we have 8 possible outcomes? What operation can I use to get an
8 using the 2 and the 4? That would be multiplication. When we’re finding the
possible number of outcomes, we can use the information in the first event and the
information in the second event and multiply those two for our outcome. This is
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called the fundamental counting principle. 2 shorts, the first event, times 4 shirts, the
second event is equal to 8 possible outcomes in our sample space.
We’re now going to be creating a foldable that we can use for sample space and
probability. We’re going to start with a piece of colored paper. We’re going to do a
hotdog fold and crease that. And then we’re going to fold that again end to end and
again end to end and crease. Open your foldable back up. Take the scissors and from
the front only cut the edge up to the fold so that you have four flaps. On the front of
the first section we’re going to include the information about sample space we just
covered. On the inside of the flap for sample space we’re going to include the
information about making a list, a tree diagram and then we’re going to make a chart
of the outcomes of heads and tails. We have a pictorial model with heads/heads,
heads/tails, tails/heads, tails/tails. And then we’re showing the fundamental
counting principle for finding the possible outcomes of flipping two coins. At the
bottom of the foldable we’re going to write the steps that you can follow to find the
sample space. You can make list, make a tree diagram, make a chart, draw a picture,
or use the fundamental counting principle. And the question is: How many outcomes
are there?
We’re now going to move to introducing probability. It is a school day. Sarah almost
always buys her lunch in the school cafeteria. How likely is it that Sarah will buy her
lunch today? Because it is something she almost always does, we’re going to say it is
likely she will buy her lunch today. Alex has a guitar less every Monday after school.
How likely is it that he will go to the basketball game on Monday after school?
Because he has a guitar lesson every Monday he has somewhere else he needs to
be. So, we’re going to say it is unlikely. All of the students who are in the 8th grade
are in school today. You are in the 8th grade. How likely is it that you are at school?
Because we’ve said that everyone in the 8th grade is at school and you are in the 8th
grade, then it is a certainty, or it is certain, that you are at school. Mrs. Johnson is at
a teacher convention out of town. How likely is it that she is at school? We’re going
to say it is going to be impossible for her to be at school because we’ve already
established that she is out of town. The words likely, unlikely, certain and impossible
are words that we use when we’re describing the probability of something. If it is a
certainty, if it is certain to happen, then the probability will be 1. If the probability is
impossible then it would be 0. If the teacher is out of town, then she can’t be at
school. The other words, likely and unlikely, describe the percentage of the
probability of something happening. If it is likely to happen, the probability will be
closer to 1. If it is unlikely the probability will be closer to 0. The probability of an
event is the measure of how likely it is that an event will occur.
We’re going to start by talking today about what a theory is. We talk about in science
a theory being something that may happen or an educated guess. Although we do
use the word theory in science we can also use it in mathematics when working with
probability. In problem 1, I’m going to take a green cube and an orange cube and
place those in the box. I have the green cube and the orange cube. I’m going to draw
a box to represent those two cubes with the G and the O. What is the likelihood of
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drawing a green? You are equally likely to draw a green as you are an orange
because there are two cubes, one is green, one is orange.
We’re now going to look at problem 2. We have two green and one orange
represented here. I’m now going to draw three boxes to represent the two green
cubes and the orange cube. The likelihood here of drawing a green would be 2 out of
3 because two of the cubes are green and only one is orange. So, what is the
likelihood of drawing that green? We’re going to say it is likely because we have two
greens and only one orange.
Let’s go back to problem 1. We have a green cube and an orange cube. Remember it
is equally likely to pull the green as it is the orange because there is one of each. We
can determine the theoretical probability of pulling the green by writing a ratio that
shows are number of favorable outcomes over the number of possible outcomes.
We’re going to write a ratio with the favorable outcome of the number of green cubes
over the total number of cubes. In this case, our total number of green cubes was 1.
Our total number of possible outcomes or cubes was 2, the green and the orange.
What does this mean? It means the probability of choosing a green cube from the
envelope is 1 out of 2 or one-half. We are working with simple probability because
we only have one event.
In our second problem, problem 2, we’re going to find what the probability of
choosing a green is. Remember, our green is our number of favorable outcomes. So,
what we’re going to do is we’re going to write a ratio for the probability of green.
That’s going to be the number of green cubes over the total number of cubes. Our
ratio then, that represents the probability of green is the number of green, remember
we have 2, over the total number of cubes, which is 3. What does this mean? It
means our probability of choosing a green is 2 out of 3 or two thirds.
What is an experiment? An experiment is some kind of test or trial. You may familiar
with the concept of doing an experiment in science but experiments can also be
applied in mathematics when we work with probability. We’re now going to be
working with experimental probability which means we’ll actually be performing an
experiment. I’m going to take an envelope and inside my envelope I’m going to place
the green cube and the orange cube that I’ve represented here in problem 1. I’m
going to reach into the envelope without looking and I’m going to pull a cube. What
I’m going to do in the third column is I’m going to make a tally mark under each color
as I pull them. My first pull was a green. I’m going to reach in here again, pull again.
My second pull is a green. I’m going to reach in here again and I have an orange.
You’re going to go ahead and pull a cube without looking 20 times. I’ve now
completed my experiment by pulling the cube from the envelope 20 times. My
outcomes were 12 green and 8 orange. I want to write this as the probability of
green. I ‘m going to use the probability of the number of times the favorable
outcomes occurred. We wanted green. That was a favorable outcome remember 12.
And the number of times I performed the experiment was 20. Because we have a
ratio that is not in simplified form we can simplify it by dividing by the greatest
common factor. Our simplified ratio then is three fifths. We’ve just determined the
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probability of choosing a green if we had a green and an orange in our envelope.
We’ve determined the experimental probability of pulling a green. That means the
next cube we pulled out of the envelope has a 3 out of 5 chance or probability of
being a green. Will we always have the same experimental probability? And the
answer is no because every experiment is going to have a different outcome.
Theoretical probability will always be the same. If you have a green and an orange,
the theoretical probability will always be 1 out of 2. The experimental probability will
change depending on how many times you’re pulling the cube. As the number of
experiments increase with experimental probability the two will become closer. Let’s
go ahead and pull 10 more cubes to see how that changes our experimental
probability. I’ve just completed 10 more trials with my experimental probability. My
total now for green is 16 and my total now for orange is 14. So green would be 16
out of 30 and orange would be 14 out of 30. This would simplify to 8 out of 15 and 7
out of 15. Remember that after we had completed 20 trials, our probability of pulling
a green was three fifths. After 30, we now have a probability of eight fifteenths,
which is very close to one half, which is what our theoretical probability is. In general,
the more trials or experiments you complete, the closer your experimental probability
is to your theoretical probability.
We’re now going to look at problem 2 where we have 3 cubes. We have 2 green and
1 orange. We’re going to place those in the envelope and we’re going to use the
same experiment that we did for problem 1. We’re going to pull a cube at a time and
keep a tally of how many times we pull orange and how many times we pull green.
Our first pull was an orange. Our second pull was a green. Orange. Green. Orange.
Green. Go ahead and complete the experiment pulling a cube 20 times. I’ve just
completed my experiment by pulling a cube 20 times. I had 12, which was my
favorable outcome out of the 20 pulls that were green. I can simplify this and I find
that three fifths of my pulls were green. The theoretical probability was two thirds
because we have 2 green and 3 total. If I chose to complete more experiments, my
experimental probability will come closer and closer to the theoretical.
We’re now going to back to the foldable that we created when we were completing
the section on sample space. The second flap of out foldable is going to have
theoretical probability. On the top flap of the theoretical probability we’re going to use
the sample of tossing a coin. When we toss a coin, there are 2 sides or possible
outcomes. So denominator is 2. One side is heads, so the numerator is 1. The
probability of heads is 1 out of 2. The information that you want to include in the
bottom half of your theoretical probability flap is: What do you think would happen
when: probability of event is the number of favorable outcomes over the total
number of possible outcomes.
The third section is going to be entitled experimental. In the section at the top of the
flap for experimental, you’re going to write the example of tossing a coin. You tossed
a coin 20 times, so the denominator is 20. It landed on heads 7 times, so the
numerator is 7. The experimental probability of heads here is 7 over 20. In the
bottom section of the experimental probability you want to write: see what happens
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when probability of the event is the number of times the event occurs over the total
number of experiments.
If we look at the word homework, this is called a compound word. It means that is
made up of more than one thing. In the word homework, we have two separate words
that have been combined to make a new word; the word home which has one
meaning, work which means another. We’re going to be doing the same thing with
probability. You have with you an envelope. We’re going to place two cubes in the
envelope. We’re going to put one green and one orange cube. You see that we have
two cubes total in the envelope. What would be the probability of drawing a green
out of this envelop? The probability of choosing a green out of my envelope would be
1 out of 2 because of the 2 cubes one of those is green. So, if I pull a green the
probability is 1 out of 2 that I can do that. If I put the first cube back, what’s the
probability that I’m going to choose an orange? It’s still going to be 1 out of 2
because one is orange and there are two cubes. Let’s look at column 1. There are 2
cubes in the envelope. We know that 1 out of the 2 cubes is green. We need to find
the probability of drawing the green. The probability we said of drawing the green is 1
out of 2 or one half. What does the word independent mean? Independent means
that one thing does not affect the other. When you work with probability,
independent events do not affect each other. Let’s look at the column labeled event
2. What did we say that our second event was? We said it was drawing an orange.
We drew the green as our first event and we drew our orange as the second event.
Because we replaced the cube after the first event, there are still 2 cubes in the
envelope. 1 out of the two cubes is orange. The probability of drawing an orange now
is 1 out of 2 or one half. The first event did not affect our second event because we
replaced the cube, the green, after the first event. As we look at the fourth column,
we see it is labeled compound probability. Compound probability means that we
combine the two separate probabilities to find the compound probability of an event.
Selecting two different cubes is compound because we have two different events.
Remember that simple probability only has one event. My possible outcomes here
would be to choose a green and then an orange. However, I could also could have
chosen green, replaced it, and drawn a green again. If I had chosen an orange first, I
could have selected the green second. But because I replaced the orange, I could
have drawn it the first time and again the second time. The probability of choosing a
green then is 2 out of the 4 or one half. The probability of choosing an orange was
also one half or 2 out of the 4. So, the probability of choosing green then orange can
be written as a multiplication, probability of green then orange. The probability of
green is one half times the probability of orange is one half and that gives us a
probability of one fourth. Let’s look back up here to see if that’s true. Of the four
outcomes, how many of then are green and orange, green then orange? First green,
second orange. And that would only be 1 out of the 4 or one fourth.
In our next example we’re going to place three cubes in an envelope, 2 greens and 1
orange. The question is: What is the probability of drawing a green, replacing it and
drawing an orange? How many total cubes do we have? We have 3. How many of
those are green? 2. So, the probability of choosing a green right now is 2 out of 3.
So, let’s say we draw the green but we replace it after we draw it. So, the probability
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of choosing and orange now is 1 out of 3. Let’s look at event 1. We know that event 1
is drawing a green. We know that there are 3 cubes in the envelope and 2 out of 3
cubes are green. So, the probability of drawing a green is 2 out of 3 or two thirds.
Now, we’re working with compound probability so you know you’re going to have
more than one event. In our second event is drawing an orange but remember we’re
going to replace the cube after the first event. So, we still have 3 cubes in the
envelope. 1 out of those 3 cubes is orange. So, our probability of drawing an orange
is 1 out of 3 or one third. Let’s move to column 4 that says compound probability. We
want to know what the probability is of choosing a green then an orange. Our
probability of choosing a green was two thirds. Our probability of choosing an orange
is one third. Remember, we multiply those two. So, our probability of choosing a
green and then and orange is two ninths.
We’re still working with compound probability and I’ve placed two cubes in the
envelope, green and an orange. This situation is: What is the probability of drawing a
green and then drawing an orange if we don’t replace the green? If I draw a green
and I don’t replace it, I only have 1 cube left instead of 2. Not replacing the first cube
affects what happens with the second event. That is what we refer to as compound
probability with dependent events. In independent probability, the second event was
not affected by the first event. There was still two to choose from. With dependent
events, what happens in event 1 affects event 2. In event 1 we have 2 cubes in our
envelope. 1 out of the 2 cubes is green. So, what is the probability of drawing a
green? The probability of drawing a green is 1 out of 2 or one half. Now, if we choose
the green, we’re only left with only 1 cube in the envelope. So, event 2 is affected by
what happened in event 1. After the first cube is picked there’s only 1 left in the
envelope. 1 out of the 1 cube is orange. The probability now of drawing an orange is
1 out of 1 or 1. How, then, do we show the compound probability of green then
orange if we do not replace it? It’s the probability of green times the probability of
orange after choosing the green. The probability of green was one half. The
probability of orange after green is 1 over 1. And when we multiply, we have a
compound probability, dependent events, of one half.
We’re now going to be looking at the second example problem. We have an envelope
with 2 greens and an orange. Three cubes are placed in an envelope. What is the
probability of choosing a green? It would be 2 out of the 3 are green. If you do not
replace the green, what would be the probability of choosing an orange? It would now
be 1 out of 2. What is the probability of our first event? It’s going to be 2 out of 3
because there are 2 greens and 3 total cubes. So, it’s 2 out of 3 or two thirds. Our
second event is going to be pulling an orange after the green is pulled. There are only
2 cubes left. 1 of those is orange. So, our probability of drawing an orange is 1 out of
2 or one half. We’re going to look at our compound probability and we know that we
have found the compound probability by multiplying the two events. The probability of
green then the probability of orange after green. Green must be our first event. The
probability of pulling a green is two thirds. The probability of pulling an orange after
the green is one half. When we multiply those two events we have two sixths, which is
simplified to one third.
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We’re going to be looking at some situations with compound probability, independent
and dependent events. Remember, independent events do not affect each other.
Dependent events means that the outcome of the second event is affected by what
happened with the first event. We have a spinner with numbers 1 through 6 and our
two events are spinning the spinner. We spin the spinner one time and then we spin
it a second time. Our question is: What is the probability of spinning a 5 and then an
even number? The probability of the first event is 1 out of 6 because on the spinner
we have the numbers 1, 2, 3, 4, 5, 6. There are 6 total outcomes but only one of
those outcomes is the number 5. On the second event, we’re going to spin again, and
there are still 6 outcomes. They want to know what the probability is of spinning an
even number. On the spinner the numbers 2, 4, and 6 would be even. So, three
outcomes would be even. The probability, then, of spinning a 5 and an even number
would be the probability of 5 times the probability of even. These are independent
events because what you spun the first time does not affect your spin the second
time. We take our probability of one sixth multiply it by the probability of the second
event which is three sixths and we have a product of 3 over 36, which can be
simplified to one twelfth. If we look at it as a dependent event we ask ourselves: what
is the probability of spinning a 5 and then an even number? Spinning a spinner is not
a dependent event because the second spin is not affected by the first spin.
Our second situation is: we have a bag that contains number cards 1 through 10.
What we want to do is draw two cards from the bag. Let’s talk about the first
example. What is the probability of drawing a 9, replacing the card and drawing and
odd number? The probability of drawing a 9 is 1 out of 10. There are 10 possible
outcomes and only one of them is a 9. The probability of drawing an odd card: there
are 10 possible outcomes and 5 of those are odd. Because we’re replacing the card,
we know that we’re going to have 10 outcomes the first event and 10 outcomes of
the second event. We’ll multiply the two probabilities and the probability of selecting
a 9, replacing the card and selecting an odd is one twentieth. In the third column we
ask ourselves: what is the probability of drawing a 9, and then drawing and odd
number without replacing the 9? This is an example of dependent events. The
probability of drawing a 9 is still 1 out of 10. At the beginning for our first event we
still have 10 cards and one of those is a 9. However, we’re not going to replace the
9, so the outcomes are 1 less, 9 possible outcomes. Four of those numbers, 1, 3, 5,
and 7 are odd, so 4 out of 9 are odd. In order to find the probability we’re going to
multiply the first event times the second event. The probability of 9, remember, is one
tenth or 1 out of 10. The probability of odd after pulling the 9, we had an outcome of
9 and 4 of those numbers were odd. We multiply the two probabilities and simplify
for a probability of 2 over 45 or two forty-fifths.
We’re now going to go back and look at the foldable we’ve been creating for sample
space and probability. The last section is going to be entitled compound. At the top of
the flap we’re going to include information on independent probability. Example:
picking a marble from 3 red, 2 blue, 4 yellow, 1 green and rolling a fair number cube.
The probability of a red and a 2 is three tenths times one sixth which is three sixtieth
which simplifies to one twentieth. Those two events are independent because the
one event does not affect the other. Underneath that you want to include information
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about dependent events. An example would be: picking two marbles from 3 red, 2
blue, 4 yellow, 1 green. The probability of red and then blue would be three tenths
times two ninths. You’re not replacing the first marble. You have a probability of 6
over 90 which simplifies to 1over 15. The formula you’re going to use to determine
independent is the probability of A and B is the probability of A times the probability
of B. The formula for dependent events is the probability of A then B, which is the
probability of A times the probability of B after A.
Let’s go back to our SOLVE problem from the beginning of the lesson. Evie is
conducting a probability experiment in her math class. She is rolling a fair number
cube and spinning a spinner which is divided into eight equal sections numbered 1
through 8. What is the probability that Evie will roll an even number on the number
cube and spin a number greater than 5 on the spinner?
The beginning of the lesson we underlined the question and we completed the
statement: this problem is asking me to find the probability that Evie will roll an even
number on the number cube and spin a number greater than 5 on the spinner?
We’re now going to move to the O step where we’re going to organize the facts. We
first start by identifying the facts. Evie is conducting a probability experiment in her
math class. That’s a fact. She is rolling a fair number cube. That’s a fact. And
spinning a spinner which is divided into eight equal sections numbered 1 through 8.
Fact. What is the probability that Evie will roll an even number on the number cube
and spin a number greater than 5 on the spinner? We’re then going to eliminate the
unnecessary facts. We don’t need to know that the experiment happens in her match
class but we do need to know she rolls a fair number cube and is spinning a spinner.
So, we list those necessary facts.
Now we’re moving to the L step. We’ll line up our plan by choosing and operation of
multiplication. We’ll then write our plan of action in words. Determine the probability
of each event. Multiply the two probabilities.
In our V step we move to estimate an answer. Our estimate is less than one half.
Then we carry out our plan. The probability of an even number: 3 out of 6 are even or
three sixths that would be 2, 4 and 6. The probability of a number greater than 5 is 6,
7, and 8. That would be 3 out of 8. We multiply those probabilities for a product of
nine forty-eighths and we simplify to three sixteenths.
We complete our SOLVE problem by examining our results. Does your answer make
sense? Compare your answer to the question. Yes, because I found the probability. Is
your answer reasonable? Compare your answer to the estimate. Yes, it is reasonable
because it is less than one half. Is your answer accurate? Check your work. The last
part of the E step is to write your answer in a complete sentence. The probability the
Evie will roll and even number on the number cube and spin a number greater than 5
on the spinner is three sixteenths.
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Let’s go back and answer the essential questions from the beginning of the lesson.
What is simple probability? Simple probability is how likely something is to occur.
What is the difference between theoretical and experimental probability? Theoretical
probability determines the probability of an event using a formula. Experimental
probability determines the probability of an event by doing and experiment. What is
compound probability? How likely that two or more events will occur. What is the
difference between independent and dependent events? In independent events, the
outcome of the first event does not affect the outcome of the second event. If events
are dependent, the outcome of the first event does affect the outcome of the second
event.
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