Download Title: Proportions

Level F Lesson 30
Simple Probability
The objective for lesson 30 is the student will investigate simple probability. We have
three essential questions that will be guiding our lesson. Number 1: What is simple
probability? Number 2: How do I use probability in everyday life? And Number 3:
What is the difference between theoretical and experimental probability?
Our solve problem for this lesson is: Davida has the following spinner. Her friend,
Eva, has a spinner with 6 numbers on it. If Davida spins the spinner, what is the
probability she will spin a 2?
We’re going to start by completing the S step for Solve. We’re going to underline the
question. What is the probability she will spin a 2? Then we’re going to complete the
statement: this problem is asking me to find the probability that Davida will spin a 2.
We’re going to be creating a foldable today that we can use to organize our
information about sample space and simple probability. We’re going to start with our
paper laid horizontally on our desk and we’re going to fold down from the top leaving
just a small space at the bottom. We’re going to fold and crease. We want to fold the
paper in thirds so we’re going to fold vertically and then bring our top down. We’re
going to open that back up. On the section that has the shorter edge, I’m going to
take some scissors and I’m going to cut to the fold. The last step I’m going to take
the flap at the bottom and I’m going to fold that up. On one of the sections I’m going
to write sample space. When you open your foldable on this part you’re going to
include information about sample space. For sample space you can: make a list,
make a tree diagram, make a chart, draw a picture, or use the Fundamental Counting
Principle. The question for sample space is: How many outcomes are there? On the
remaining space you’re going to include an explanation of sample space. There’s a
list and what we’ve done is create this list and the tree diagram using the possible
outcomes from flipping a coin. If we have two coins to flip we could have a: head and
a head, or a head and a tail or we could have a tail and a head, and a tail and a tail.
So if our first outcome was either heads or tails each of those would have 2
outcomes. Here’s a chart we’ve created showing those outcomes and we’ve drawn a
picture. The last thing we’ve done is explain it using the Fundamental Counting
Principle. There are 2 possible outcomes for our first event, 2 for the second and
when we multiply those, we have a sample space of 4.
We’re going to begin by discussing some of the vocabulary that we use with
probability. I have four words on my chart here that are some of the things we may
say when we’re talking about probability. We have likely events, unlikely events,
certain and impossible. When we think about situations, we can categorize them in
one of those four ways. Let’s take a look at likely. If you are school, what is the
probability that you will eat at lunch at school on a school day? That’s a likely event
because it will probably happen. Most of the students who are at school will eat their
lunch. Is it likely or unlikely that you will stay overnight at school? It’s probably an
unlikely event because only in a very rare case, such as a special event for a club,
1
would you ever be staying overnight at school. So you would probably not stay
overnight. That’s an unlikely event. Is it certain that the sun will rise tomorrow? This
is probably a certain statement as the sun rises each day even when we don’t see it.
How about an impossible event? The temperature outside today will be 150 degrees.
It’s probably impossible because that temperature does not occur here on earth. You
may think of some other events and discuss those with your teacher about things
that would be likely, unlikely, impossible and certain.
We’re going to be talking about simple probability with a fair number cube. On our
page we have an example of a fair number cube that would have the number 1
through 6 on the sides. There are six possible outcomes if we were to toss that
number cube: 1,2,3,4,5, and 6. Each of those outcomes is equally likely or has a fair
chance of being tossed. If the number cube is tossed it has an equal probability of
landing on each of the sides.
In example 1, we’ve drawn a picture of each of the possible outcomes for the fair
number cube toss. What is the probability of tossing a 4 on the fair number cube? If
we were to toss the fair number cube, this is the 6 possible rolls we could have, it
could be a 1, a 2, a 3, a 4, a 5, or a 6. However, only one of those rolls is a 4. So,
there is one 4 in the possible rolls. The probability of rolling a 4 is 1 out of 6. The
probability can be written as 1 to 6, 1 colon 6 or one-sixth with a fraction bar.
In example 2, we’re using the same sample space and our question is: what is the
probability of tossing and even number on the number cube? Out of our choices, 3 of
those numbers are even. 2, 4 and 6 are even numbers. The probability of rolling an
even number is 3 out of the 6. This probability can be written as 3 to 6, 3 colon 6, or
the fraction 3 over 6. This ratio of three-sixths can be simplified to one-half.
In example 3, we still have our same number of possible outcomes and our question
is: What is the probability of tossing a number greater than 2. There are 4 numbers in
our sample space, 3, 4, 5, and 6 that are possible rolls that are greater than 2. The
probability of rolling a number greater than 2 then is 4 out of the 6. The probability
can be written as 4 to 6, 4 colon 6, or the ratio four-sixths. The ratio of 4 over 6 can
be simplified to 2 over 3, or two-thirds.
We’re going to take a look at simple probability theoretical. We’re going to start. Our
problem is we’ve been give two cubes of different colors. We have a green and a red.
So the first thing I’m going to do is I’m going to take my green marker and shade the
representation of that and I also have a red cube. When we have a theoretical
probability it says: What is the probability of drawing one color. All the possible colors
are called the possible outcomes. In this case there are 2 possible outcomes: green
and red. If I were to pick one of these cubes without looking I would choose either a
green or a red. If I were to predict green then the event would be: choosing green. If
you predict red, then your event would be to choose red. There are two possible
outcomes: green and red. This theoretical probability of an event is a ratio written as
the number of favorable outcomes over the number of probable outcomes. In the
case of our example, the probability of choosing green, which is the event, is written
2
as a ratio as the number of green cubes we have in the envelop, which was 1, over
the total number of cubes, or one-half: 1 green and 2 green and red. What does this
mean? It means the probability of choosing a green cube from the envelope is 1 out
of 2.
In our second example, we’re going to have 3 cubes: 2 of one color and 1 of the
second color. So, we have 2 reds and 1 is green and we’re trying to find the
probability of drawing the first color. First thing I’m going to do is shade in to
represent a pictorial example of my three number cubes. If I put my 3 cubes in an
envelope, 2 reds and a green, what color will I pull out if I don’t look? I would pull our
either a red or a green. I would be more likely to pull out a red because there are 2
reds in the envelope and only 1 green. If you predict I will pull a red out of the
envelope then the event would be choosing red. If you predicted green then your
event would be choosing green. There are still only 3 possible outcomes: red, red, or
green. So, we’re going to use our formula for theoretical probability to determine the
probability of choosing a red cube. The probability of red is the number of red cubes
over the total number of cubes. We have 2 red in the envelope and a total of 3
cubes. What does this mean? The probability of choosing a red cube from the
envelope is 2 out of 3.
We’re going to move on and talk about experimental probability. I’ve left my
information from theoretical probability but we’re going to talk about experimental
probability. Experimental probability is where we’re going to actually have to pull the
cubes from the envelope and base our probability on the results of our experiment.
We still have a green and a red cube. We’re going to shade those in and if we place
those cubes in an envelope we’re going to do an experiment. What you’re going to do
is, with your partner, you’re going to put the cubes in and without looking draw a
cube. You’re going to create a tally chart here in this box that indicates how many
times you drew each color. You’re going to do 20 events or 20 tries of pulling a cube
with your partner. Ok, I have completed the experiment by pulling a cube without
looking 20 times from my envelope. Remember, your results will differ from mine
because each experiment will be different. I had 8 times that I pulled a red and I had
12 times that I pulled a green. What’s the difference between the experimental and
the theoretical probability? In the theoretical probability, we base our probability on
the favorable outcomes over the possible outcomes. In the experimental probability
we base it on the favorable outcomes over the number of trials in our experiment. In
the case of my experiment, we said green; I had 12 times that I drew a green out of
20. That can be simplified to 3 over 5. The question is: why is my experimental
probability not the same as my theoretical? What happens is if the number of
experiments is increased; the experimental probability will get closer to the
theoretical probability.
The simple probability example problem that we’re going to use to talk about
theoretical and experimental probability is this: Given the numbers 2, 3 and 4, what
is the probability of getting a 3? I want you to take a simple index card and we’re
going to cut that into 3 sections. After we do that, we’re going to number the
sections: 2, 3 and 4. Those numbers then represent our sample space. To determine
3
the theoretical probability of drawing a 3 remember we have our 3 number cards in
our envelope: a 2, 3 and 4. There is only one 3 in that problem. So, the number of
3’s is 1 and the total number of numbers that are possible to draw is 3. What does
this mean? It means that the probability of choosing a 3 from the envelope is 1 out of
3. For our experimental probability we have the number of times that a favorable
outcome occurs over the number of trials in the experiment. We’re going to complete
20 trials again and we’re going to draw the number cards without looking 20 times.
I’ve just finished drawing my number cards, 20 separate trials or experiments, and
I’ve drawn a 3 seven times. That means my experimental probability of drawing a 3 is
7 out of 20.
We’re going to be completing the foldable for probability. We’ve already done the
sample space section. Now we need to label the middle section theoretical
probability and the third section experimental probability. When I open my theoretical
probability flap, on this section underneath the title, we’re going to include this
information: What do you think would happen when…? And this is our formula for the
probability. The event is equal to the number of favorable outcomes over the total
number of possible outcomes. On the inside of the theoretical probability folder
you’re going to include an example of tossing a coin. There are 2 sides which are
possible outcomes. So, the denominator is 2. One side is heads, so the numerator is
1. The probability of heads is one-half.
In the last section of the probability foldable, underneath the flap where you have
written experimental probability you’re going to write this: See what happens when
you do it! To remind to that in experimental probability you need to perform an
experiment. The probability of an event is the number of times the event occurs over
the total number of experiments. In the last part of your experimental probability
foldable you’re going to include this information. The example is tossing a coin. If you
tossed a coin 20 times, your denominator is 20. If you landed on heads 7 times, your
numerator is 7. Your probability of heads in this experimental probability was 7 over
20.
We’re now going back to the solve problem from the beginning of the lesson. Davida
has the following spinner. Her friend, Eva, has a spinner with 6 numbers on it. If
Davida spins the spinner, what is the probability she will spin a 2? We’ve already
completed the S step by underlining the question and completing the statement: the
problem is asking me to find the probability that Davida will spin a 2.
We’re going to move to the O step and we’re going to identify the facts. Davida has
the following spinner. Her friend, Eva, has a spinner with 6 numbers on it. If Davida
spins the spinner, what is the probability she will spin a 2? Then we’re going to
eliminate the unnecessary facts. We have a picture of the spinner and we know she
has that. Her friend, Eva, has a spinner with 6 numbers on it. We don’t need to know
that either. Now we’re going to list the necessary facts. The spinner that she has has
4 sections labeled 1, 2, 3, and 2. Davida will spin one time and her event is a 2.
Now we’re going to line up our plan. We’re going to choose an operation, which is N/A
4
meaning not applicable for this problem. Write in words what your plan of action will
be. We’re going to determine the favorable outcomes and the possible outcomes and
we’re using the formula for theoretical probability. The probability of the event is the
number of favorable outcomes over the number of possible outcomes.
Now we’re going to make an estimate. Our estimate is about 2 out of 4. Then for V,
we’re going to carry out our plan. Our favorable outcomes are 2. Our possible
outcomes are 4. The probability of spinning a 2 is 2 out of 4 or 2 out of 4 or one-half.
We’re going to move to the E step now. Does your answer make sense? Yes, because
we found the probability of spinning a 2. Is your answer reasonable? Yes, because 1
out of 2 or 1 over 2, one-half matches my estimate of 2 out of 4 or two-fourths. Is
your answer accurate? Yes. Write your answer in a complete sentence. The
probability that Davida will spin a 2 is 2 out of 4, or two-fourths, which can be
simplified to one-half.
Now we’re going to go back and answer the essential question from the beginning of
our lesson. What is simple probability? How likely something is to occur. How do I use
probability in everyday life? Prediction of the weather may be one possibility and the
other examples that were discussed at the beginning of your lesson. 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 an experiment.
5