12.6: Introduction to Probability
Introduction
The Spinner Game
At the amusement park, Keith and Trevor went over to the carnival booths to try their luck at a few games. Keith played one game of “Whack a Mole” and won a ticket for an ice cream cone. Trevor threw a golf ball into a fish bowl and won a gold fish.
Then they both moved on to games of chance. After checking out several different games, they decided to play a game with a spinner. In this game you spin the spinner and whichever number you get determines the number of chances that you have. The object of the game is to use a bow and arrow to hit a target. In the autumn, the sixth grade had learned some archery and Keith had been particularly good at it.
“You’ve got this,” Trevor said supporting Keith. “You were the best one in the class.”
“Yes, but I want to spin the highest number on the spinner that I can.”
Keith and Trevor looked at the spinner. There were 10 sections on the spinner. That means that Kevin could spin anywhere from a one to a 10. If he only spun a one, then he would only get one shot at the target. If he spun a ten, then he would get 10 chances.
Spinner courtesy of http://etc.usf.edu/clipart/37700/37714/spinner-10_37714.htm
“What do you think my chances are of spinning an 8, 9 or 10?” Kevin asked Trevor.
“I don’t know, let me think about that. I also wonder what that chance would be as a percentage.” Trevor said.
“More importantly, what are the chances that I won’t spin an 8, 9, or 10?” Kevin mused.
While Kevin and Trevor do their own figuring, it is time for you to learn about probability. At the end of this lesson, we will return to this problem and you can help Trevor and Kevin figure out the probability.
What You Will Learn
In this lesson, you will develop an understanding of the following things related to probability.
- Recognize the probability of an event as the ratio of favorable outcomes to possible outcomes.
- Describe probabilities of events as fractions, decimals or percents.
- Find the probabilities of complementary events.
- Predict whether specified events are impossible, unlikely, likely or certain.
Teaching Time
I. Recognize the Probability of an Event as the Ratio of Favorable Outcomes to Possible Outcomes
As you could see in the introduction problem, Kevin and Trevor are working on figuring out “chances.” When we figure out the chances of something happening or not happening, we say that we are figuring out the probability or the chances of an event happening.
We use probability all the time in real life situations. If you watch the weather in the morning you may hear the meteorologist talk about a 20% chance or rain or snow. In this case a percentage gives us the probability that it would rain. While there is a 20% chance that it will rain, there is an 80% chance that it won’t rain. All in all, we are still talking about probability.
How can we calculate probability?
To calculate probability we use a ratio. If you remember back to earlier lessons, you will remember that a ratio is a way of comparing two quantities. With probability, we can compare the number of favorable outcomes to the amount of possible outcomes.
Here is our ratio.
\begin{align*}P = \frac{\# \ of \ Favorable \ Outcomes}{\# \ of \ Possible \ Outcomes}\end{align*}
Notice that the ratio is in fraction form. That is one way that we can compare to figure out the probability of an event happening.
How can we apply this ratio?
To apply this ratio, we have to look at an example. As you read this example, think about the number of possible outcomes first. That is our denominator. Then go to the number of favorable outcomes.
Let’s look at an example.
Example
Mark is rolling a number cube that is numbered 1 – 6. What are the chances that Mark will roll a 2?
To work through this problem and figure out the probability we first need to determine the number of possible outcomes. Since the number cube is numbered 1 – 6, there are only 6 possible outcomes. That is our denominator.
\begin{align*}P = \frac{number \ of \ favorable \ outcomes}{6}\end{align*}
Next we think of the number of favorable outcomes. Since we are only looking for a two, there is one favorable outcome. That is our numerator.
\begin{align*}P = \frac{1}{6}\end{align*}
That one was an introductory problem. Now let’s look at one that is a little more complicated.
Example
Jessie spins the same number cube. She wants to spin an odd number. What are the chances that she will spin an odd number?
Let’s break this one down. First, the number of possible outcomes did not change. It is still a 6.
\begin{align*}P = \frac{\# \ of \ Favorable \ outcomes}{6}\end{align*}
The number of favorable outcomes did change. We are looking for an odd number. If we count from 1 – 6, there are three odd numbers. Therefore, the number of favorable outcomes is 3.
\begin{align*}P = \frac{3}{6} \ or \ \frac{1}{2}\end{align*}
Notice that we can simplify the probability too. Sometimes that will give an even clearer picture of the likelihood that the event will or will not happen.
Practice finding probability. Write a ratio to show the probability for each question below.
Jake put eight colored squares into a bag. There are two reds, four yellows, one green and one blue.
- What is the probability of Jake pulling out a red cube?
- What is the probability of Jake pulling out a yellow cube?
- What is the probability of Jake pulling out a yellow or blue cube?
Take a few minutes to check your work with a friend. Did you simplify when possible?
II. Describe Probabilities of Events as Fractions, Decimals or Percents
In the last section we looked at probability as a ratio in fraction form.
\begin{align*}P = \frac{\# \ of \ Favorable \ Outcomes}{\# \ of \ Possible \ Outcomes}\end{align*}
We wrote our ratios as fractions and simplified them when we could.
Example
\begin{align*}\frac{3}{6}=\frac{1}{2}\end{align*}
Let’s think about fractions for a minute. Fractions mean a part of a whole. Decimals and percents also mean a part of a whole. Therefore, we can write our probabilities as fractions, but we can also write them as decimals or as percents.
Let’s practice writing the following probabilities three different ways.
Example
A bag has four cubes in it, a red cube, two yellow cubes and one blue cube. What is the probability of drawing a red or yellow cube out of the bag?
To work on this problem, let’s first write a fraction to show the ratio of possible outcomes and favorable outcomes. There are four cubes in the bag, so there are four possible outcomes. This is our denominator.
\begin{align*}P & = \frac{\# \ of \ favorable \ outcomes}{\# \ of \ possible \ outcomes}\\ P &= \frac{\Box}{4}\end{align*}
Next, we need to figure out the favorable outcomes. We want a red or a yellow. There are two yellow cubes and one red cube. That means that there are three favorable outcomes.
\begin{align*}P = \frac{3}{4}\end{align*}
Our next step is to write this as a decimal. To write \begin{align*}\frac{3}{4}\end{align*} as a decimal, we need to convert the fraction to one with a denominator that is a multiple of ten. We can create a proportion, or equal fraction with a denominator out of 100 to do this.
\begin{align*}P = \frac{3}{4} & = \frac{\Box}{100}\\ 4 \times 25 &= 100\\ 3 \times 25 &= 75\\ P &= \frac{75}{100} \ or \ .75\end{align*}
Now we can take the decimal and make it a percentage. If you look at the fraction out of 100 it is already clear what the percentage is. The percentage is 75% because percent means out of 100.
If you were working with the decimal only, then you move the decimal point two places to the right and then add the % sign. You move it two places because that is hundredths and % means out of 100.
Practice writing the following probabilities from the first set of exercises and write each as a decimal and a percentage.
- \begin{align*}\frac{1}{4}\end{align*}
- \begin{align*}\frac{1}{2}\end{align*}
- \begin{align*}\frac{5}{8}\end{align*}
Take a few minutes to check your work. The last one was tricky. Did you get it correct? Fix any errors before moving on.
III. Find the Probabilities of Complementary Events
What happens when we know the likelihood that something will happen? Well, we can determine or base our actions on that event happening.
If there is a 10% chance of rain, what is the probability that it will be sunny? We can say that there is a 90% chance that it will be sunny.
What are the chances that it will not be sunny?
If someone only knew that there was a 10% chance that it would be rainy, and that if it wasn't rainy the only other option was to be sunny, could they tell the chance of it being sunny? To figure this out, we have to figure out what the chances are of something not happening. This is called a complementary event.
If there is a 10% chance that it will not be sunny, then there is a 90% chance that it will be sunny.
Write the complementary event for the probability in this example.
Example
There is a 50% chance that Mary will be coming over on Saturday.
To write the complementary event, we look at the opposite probability. There is a 50% chance that Mary will be coming over, so there is a 50% chance that she will not be coming over.
There is a 50% chance that Mary will not be coming over.
We can write complementary events as fractions, decimals and percents. Use whatever form is used in the example and have the complementary event match that form.
Practice writing a complementary event for each example.
- There is a 20% chance that it will snow tonight.
- There is a 55% chance that we will win the football game Friday night.
Take a few minutes to check your work.
IV. Predict Whether Specified Events are Impossible, Unlikely, Likely or Certain
We can also predict how likely an event is to happen or not to happen based on common sense. Some things we can know for certain and some things are left up to chance.
The sunrise is one of those events. We know that the sun will rise tomorrow. Sometimes we won’t see it due to weather, but it certainly will rise.
You can also catch yourself arguing about this too. How do we know that it will rise? You could find yourself debating this with a friend for a long time. However, we need to use common sense when we are thinking about these things and not just figuring the probability using numbers.
Don’t get too caught up!
Predict whether each event is likely, impossible, unlikely or certain.
- The team lost its last four games, it is __________ that they will win tonight.
- On her fifth birthday, Joanna turned five years old.
- A car can fly.
Take a few minutes to discuss your answers.
Real Life Example Completed
The Spinner Game
Here is the original problem once again. Reread this problem and underline the important information. Then use what you have learned about probability to help Keith figure out his chances.
At the amusement park, Keith and Trevor went over to the carnival booths to try their luck at a few games. Keith played one game of “Whack a Mole” and won a ticket for an ice cream cone. Trevor threw a golf ball into a fish bowl and won a gold fish.
Then they both moved on to games of chance. After checking out several different games, they decided to play a game with a spinner. In this game you spin the spinner and whichever number you get determines the number of chances that you have. The object of the game is to use a bow and arrow to hit a target. In the autumn, the sixth grade had learned some archery and Keith had been particularly good at it.
“You’ve got this,” Trevor said supporting Keith. “You were the best one in the class.”
“Yes, but I want to spin the highest number on the spinner that I can.”
Keith and Trevor looked at the spinner. There were 10 sections on the spinner. That means that Keith could spin anywhere from a one to a 10. If he only spun a one, then he would only get one shot at the target. If he spun a ten, then he would get 10 chances.
Spinner courtesy of http://etc.usf.edu/clipart/37700/37714/spinner-10_37714.htm
“What do you think my chances are of spinning an 8, 9 or 10?” Keith asked Trevor.
“I don’t know, let me think about that. I also wonder what that chance would be as a percentage.” Trevor said.
“More importantly, what are the chances that I won’t spin an 8, 9, or 10?” Keith mused.
First, let’s figure out what the probability is that Keith spin an 8, 9, or 10. There are 10 sections on the spinner. 10 is the denominator because it is the total number of possible outcomes.
8, 9, or 10 is the numerator. There are three favorable outcomes.
\begin{align*}P = \frac{3}{10}\end{align*}
Next, Trevor wondered what that chance would be if written as a percentage. To figure this out, we need to convert the fraction to a percent. We do this by creating an equal fraction out of 100.
\begin{align*}\frac{3}{10}= \frac{30}{100}\end{align*}
Keith has a 30% chance of spinning an 8, 9 or 10.
What about the chances of not spinning an 8, 9 or 10?
Well, if there is a 30% chance that he will spin one of those numbers, then there is 70% chance that he won’t.
You could say that the odds are against Keith spinning one of those numbers. It is unlikely that he will do so given his chances. However you never know.
Keith spins the spinner and spins a 6. He didn’t get the 8, 9 or 10, but he is happy with 6 chances. He warms up and aims the arrow. On the fifth try, Keith hits the bullseye of the target. He chooses a stuffed pink giraffe to take home to his sister.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Probability
- the chances that something will happen. It can be written as a fraction, decimal or percent.
- Ratio
- compares two quantities. In probability the ratio compares the number of favorable outcomes to the number of possible outcomes
- Complementary Events
- For every probability that something will happen, there is a probability that it won’t happen. These two ratios are complementary events.
Technology Integration
Khan Academy, Basic Probability
James Sousa, Introduction to Probability
Time to Practice
Directions: A bag has the following 10 colored stones in it. There are 2 red ones, 2 blue ones, 3 green ones, 1 orange one, and 2 purple ones. Write a fraction to show the following probabilities.
1. One orange stone
2. A red stone
3. A green stone
4. A yellow stone
5. A blue stone or an orange one
6. A red one or a blue one
7. A green one or an orange one
8. A blue one or a green one
9. A blue one or a purple one
10. A purple one or a red one
Directions: Write each fractional probability in numbers 1 – 10 as a decimal and a percent.
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Directions: Use common sense and make a prediction, use likely, impossible unlikely or certain to describe each statement.
21. Our team has a perfect record. It is _________ that we will win on Saturday.
22. A baby born will either be a boy or a girl.
23. A pig will fly through the sky.
24. A cat will like a dog.
25. There is an 85% chance it will rain. It is ________ that it will rain.