Download G 29 Simple Probability

Level G Lesson 29
Simple Probability
In lesson 29 the objective is the student will investigate simple probability. We have
3 essential questions that will be guiding the 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?
The solve problem for our lesson: Josh and Daniel are playing a probability game in
math class. Each of the boys rolls a fair number cube with the numbers 1, 2,3,4,5
and 6 on the cube. Josh is trying to roll a number less than 3. Daniel is trying to roll
an even number. If they each roll the number cube one time which student is more
likely to roll one of their numbers?
We are going to study the problem and we are going to start by underlining the
question, which student is more likely to roll one of their numbers? We are going to
complete the statement this problem is asking me to find which student is more
likely to roll one of their numbers.
We are going to be creating a foldable that we can use for the concepts of probability.
We are going to start by placing the sheet on your desk horizontally so it is wider
across the bottom. We are going to take the top of the paper and fold it down almost
to the bottom of the page. We are going to leave about one-half inch at the bottom;
we are going to crease it on the fold at the top. Then I am going to turn my paper
vertically. After we turn our paper vertically we are going to fold it in half, crease on
the fold and then fold it in half again. When we open up the foldable we have 4
sections. You are going to take your scissors and cut on the fold just on the top
section. You have now created a foldable that we are going to use for probability. On
the bottom flap you are going to flip that up and you can write the word sample space
on the bottom flap of one of the four sections. Inside the top flap you are going to
write this information about sample space. Make a list. Make a tree diagram. Make
a chart. Draw a picture. Or use the fundamental counting principle. And the
question we are going to be looking for when we find sample space is how many
outcomes there are. In the bottom section of your sample space we are going to
create a list and a tree diagram using the idea of flipping two coins. We are going to
look at the possible outcomes for flipping a coin twice. If the first flip is a head the
second could be a head or if the first flip is a head the second could be a tail. We
could also have tails, heads, tails, tails. We have created that using a tree diagram.
We have also charted those outcomes how many times they would occur. We have
created a picture showing those and then we have demonstrated how to use the
fundamental counting principle to find the number of possible outcomes.
We are 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 that we
may say when we are 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 at school what
is the probability that you will eat lunch at school on a school day. That is a likely
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event because it will probably happen. Most of the students at school will eat their
lunch. Is it likely or unlikely that you will stay overnight at school. It is probably an
unlikely event because only in a very rare case such as a special event for a club
would you ever be staying overnight at school. So you would probably not stay over
night at school. That is 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 is 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, certain, possible and impossible.
In example 1 the question is what is the probability of rolling a 4 on the fair number
cube? Or tossing a 4 on the fair number cube? The 4 is the favorable outcome we
are looking for. In order to write the probability of a 4, we need to have the
relationship between the desired outcome or the favorable outcome and the possible
outcomes. So the probability of rolling a 4 is 1 out of 6. The probability can be
written; 1 out of 6. 1:6 or the ratio one-sixth.
In Example 2 our question is what is the probability of tossing an even number on the
number cube? There are 3 possible outcomes in this example. 3 of the values are
even numbers (2, 4, and 6). 2, 4, and 6 are the favorable outcomes for what we are
looking for. In order to write the probability of tossing an even number you write the
relationship between the favorable outcomes and the number of possible outcomes.
So our relationship between favorable and possible is 3 to 6, that can be written also
as 3 to 6, 3:6, or 3 over 6. The ratio of 3 over 6 can be simplified to 1 over 2 or onehalf.
In example 3: the question is what is the probability of tossing a number greater
than 2? How many of the possible outcomes are greater than 2? There are 1,2,3,4
and we listed those down here 3, 4, 5 and 6. So out of the possible outcomes of 6, 4
of those are favorable outcomes. In order to write the probability of tossing a number
greater than 2 we write the relationship between the favorable outcome and possible
outcomes. The can be written as 4 to 6, 4:6, or the ratio of 4 over 6. The ratio of 4
over 6 can be simplified as two-thirds.
We’re going to be working with simple probability theoretical. The problem is we are
given 2 cubes in different colors. We have a blue cube and a red cube we want to
find the probability of drawing a blue. In Theoretical probability remember we had to
find the relationship between the favorable outcomes and the possible outcomes.
Our favorable outcome is the number of blue cubes. The possible outcome is the
total number of cubes. We are going to write that as a ratio. One of the cubes is blue
that is our favorable outcome. There are two possible outcomes blue and red. What
does this mean? The probability of choosing a blue cube from the envelope is 1 out
of 2.
In the next problem we have 3 cubes, 2 of one color and 1 of the second color. We
have 2 reds and a blue, we want to know what the probability is of choosing one of
these reds. Again we have our favorable outcomes in relationship to the possible
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outcomes. So the probability of choosing a red is a ratio the number of red cubes
which is 2 in relationship to the total number of cubes. There are 2 red and there are
3 total cubes. What does this mean that the probability of choosing a red cube from
the envelope is 2 out of 3.
We are now going to talk about experimental probability with simple probability. We
are going to be given 2 cubes again of different colors blue and red. When we did
theoretical probability we predicted that one out of the 2 cubes was blue. That meant
that the probability of choosing a blue would be 1 out of 2. We are going to take a
red and blue cube and put in the envelope. What we are going to do is do an
experiment with those cubes. We are going to without looking reach into the
envelope and take out a cube.
When we do that we are going to mark that selection on our page, and make tally
marks for each draw. After each draw we are going to return it to the envelope and
select again without looking. You’re going to repeat this process until you have drawn
20 times, I have now completed my 20 draws out of envelope I drew a red cube 8
times and I drew a blue cube 12. There is a formula that we can use to show
experimental probability. We have the probability of an event, is the number of times
the favorable outcome occurs, over the number of trials in the experiment. For our
example the probability of blue was the number of times our favorable outcome
occurs which was 12 over our number of trials which was 20. Is my theoretical
probability close to my experimental? We can also simplify this to help us compare
the two. So my theoretical was 1 out of 2 or one-half. My experimental of blue was 3
over 5. The probability that’s experimental is not always going to be the same as
your theoretical. The more tries or the more experiments you perform the more likely
that your probability that’s experimental will be closer to the theoretical probability.
For this problem we are going to be using numbers 1, 2, and 3. We want to know
what the probability of getting a 3 is. For the theoretical probability we are going to
find the number of favorable outcomes over the number of possible outcomes. Our
favorable outcomes are the number of 3’s in the problem over the total number of
numbers. In this problem if we have a 1, 2, and a 3, there is only one of those values
that is a 3. However there are 3 numbers. What does this mean? The probability of
choosing a 3 from the envelope is 1 out of 3.
Let’s move over to the experimental probability. In order to find the experimental
probability we are going to have to do an experiment. We are going to take an index
card and cut it into 3 pieces. On each of the pieces you are going to write the value
of 1, 2, and 3. You are going to place those 3 number cards into your envelope. This
is the same experiment we did in the sample problem except instead of using cubes
we are using number cards, 1, 2, and 3. What I am going to do is I am going to draw
from the envelope without looking. I am going to create a tally chart to show each
number as I pull it. I replace it, and without looking I draw again. I have now
completed 20 draws from my envelope containing the number cards. I have 6, 7,
and 7 for a total of 20 draws. The number of times my favorable outcome occurs, I
have to go back and my favorable outcome was a 3. So the probability of a 3 is 7. My
number of trials is 20. The experimental probability for my experiment was 7 to 20 or
the ratio 7 over 20.
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At the beginning of the lesson we created a foldable to use for probability. We
completed the sample space section at that time. Now we are going to complete the
theoretical and experimental sections. Write the word theoretical on one flap and the
word experimental on the next flap. Inside the flap for our theoretical we are going to
write what do you think would happen when and the probability formula for
theoretical is, the probability of an event, is equal to the number of a favorable
outcomes over the total number of possible outcomes. The example you are going to
write inside your section on theoretical probability is this: tossing a coin, there are 2
sides, or possible outcomes so your denominator is 2. One side is heads, so the
numerator is 1. The probability of flipping a coin and having it landing on a head is 1
out of 2. Inside the flap for experimental probability you are going to write, see what
happens when you do it. That’s to remind you that in experimental probability there
needs to be an experiment that occurs.
The formula we use for the experimental probability is the probability of an event is
equal to the number of times the event occurs over the total number of experiments.
The example we are going to include inside of our experimental flap is tossing a coin.
We tossed a coin 20 times so the denominator is 20. We landed on the head 7 times
so the numerator is 7. The probability of heads in this experiment was 7 over 20 or 7
to 20.
Lets go back to the SOLVE problem at the beginning of the lesson. Josh and Daniel
are playing a probability game in math class. Each of the boys rolls a fair number
cube with the numbers 1, 2,3,4,5 and 6 on the cube. Josh is trying to roll a number
less than 3. Daniel is trying to roll an even number. If they each roll the number
cube one time which student is more likely to roll one of their numbers? We studied
the problem by underlining the question and completing the statement this problem
is asking me to find, which student is more likely to roll one of their numbers.
I am going to organize my facts and I am going to start by identifying the facts. Josh
and Daniel are playing a probability game in math class. That’s a fact. Each of the
boys rolls a fair number cube with the numbers 1, 2,3,4,5, and 6 on the cube, my
next fact. Josh is trying to roll a number less than 3. That is another fact. Daniel is
trying to roll an even number, that’s another fact. After we have identified the facts
now we are going to eliminate the unnecessary facts. Josh and Daniel are playing a
probability game in math class. That fact is not necessary to find out which student is
more likely to roll one of their numbers. However each of the remaining facts is
necessary. We need to know they are rolling a number cube and what the sides are
labeled. We need to know Josh is trying to roll a number less than 3 and we need to
know that Daniel is trying to roll an even number. We also need to know that they roll
one time. So we have all that information listed under necessary facts.
We are now going to move to the L step where we are going to line up our plan. We
don’t have an operation because we are finding probability. Our plan of action will be
to determine the favorable outcomes and possible outcomes for each of the boys.
Then we are going to compare the probabilities. We are going to use our formula for
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theoretical probability. Our formula for theoretical probability is the probability of the
event is the number of favorable outcomes over the number of possible outcomes.
My estimate is Daniel. Then I am going to carry out my plan. My favorable outcomes
for Josh are 2. Because he is trying to role a number less than 3. On our number
cube if we have a 1, 2, 3, 4, 5, and 6 there were only 2 numbers that are less than 3.
The possible outcomes for Josh were 6. So the probability of a number less than 3 is
2 out of 6 or two-sixths which can be simplified to one-third. The favorable outcomes
for Daniel are 3 remember he was trying to roll an even number, so his favorable
outcomes would be 2, 4, and 6. The probability of rolling an even number is 3 out of
6 or three-sixths which can be simplified to one-half. One–half is greater than onethird. Because one-half is greater than one-third then my answer is going to be
Daniel.
Does your answer make sense? Yes because I found that Daniel is more likely to roll
his number. Is your answer reasonable? I am going to compare it to my estimate. I
actually estimated that Daniel would be more likely to roll his number. Is your answer
accurate? I am going to go back and check my work. Write your answer in a complete
sentence. Daniel is more likely to roll one of his numbers.
Now we are going to go back and answer the essential questions from the beginning
of our lesson. What is simple probability? How likely something is to occur. How do I
use probability in every day life? One of the possibilities here is to predict the
weather. There may be 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.
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