12.1: Theoretical Probability
Introduction
The Talent Show
Since the opening of J.S. Middle School, a tradition has been the end of the year talent show. The school opened ten years ago, and within that time there have been 8 talent shows. There were two years when the school was not able to host one because there was flooding or repairs were being done in auditorium.
“I wonder if we are going to have the talent show this year,” Carmen asked at lunch one day.
“I am sure that we are,” Tyler said, biting into his ham sandwich. “After all, there were only two years that the talent show did not happen and that was because of the circumstances.”
“Well, are there any circumstances this year?”
“I don’t think so. The probability is high that it is going to happen.”
“What is the probability of the talent show happening?” Carmen asked, taking a sip of milk.
Before Tyler answers, let’s think about this. What is the probability of the event happening?
To answer this question, you need to know about probability. This lesson will teach you how to figure out probability by thinking about favorable outcomes and total outcomes. By the end of the lesson, you will know how to answer this question.
What You Will Learn
In this lesson, you will learn how to complete the following:
- Recognize the theoretical probability of an event as the ratio of favorable outcomes to possible outcomes.
- Calculate simple theoretical probabilities.
- Write theoretical probabilities as fractions, decimals or percents.
- Make predictions based on theoretical probabilities.
Teaching Time
I. Recognize the Theoretical Probability of an Event as the Ratio of Favorable Outcomes to Possible Outcomes
This chapter is all about probability. Probability is something that you hear about all the time. Anytime you talk about the chances that something will or won’t happen, you are talking about probability. The trick about probability is that it isn’t just about "educated guessing". It is also about math. There are mathematical ways of figuring out the likelihood that an event is going to or not going to occur. But let’s start with an understanding of probability.
What is probability?
Probability is the likelihood that an event will occur. It is a mathematical way of calculating how likely an event is likely to occur. An event is a result of an experiment or activity that might include such things as:
- flipping a coin
- spinning a spinner
- rolling a number cube
- choosing an item from a jar or bag
How do we calculate probability?
We calculate probability by looking at the ratio of favorable outcomes to total outcomes in a given situation. In ratio form, the probability of an event is:
\begin{align*}P (\text{event}) = \text{favorable outcomes} : \text{total outcomes}\end{align*}
Write this ratio down in your notebook.
An outcome is a possible result of some event occurring. For example, when you flip a coin, “heads” is one outcome; tails is a second outcome. Total outcomes are computed simply by counting all possible outcomes.
Chapter Note: Keep in mind as you go through this chapter that all outcomes used are presumed to be “fair.” That is – when you flip a coin, the outcomes of heads or tails are equally likely. When you spin a spinner, sections are all of equal size and equally likely to be landed on. When you toss a number cube, faces of the cube are the same size and again equally likely to be landed on. And so on.
For flipping a coin:
\begin{align*}\text{total outcomes} & = \text{heads, tails}\\
& = 2 \ \text{total outcomes}\end{align*}
For tossing a number cube:
\begin{align*}\text{total outcomes} & = \cdot \ 1 \ \cdot \cdot \ 2 \ \cdot \cdot \cdot \ 3 \ \cdot \cdot \cdot \cdot \ 4 \ \cdot \cdot \cdot \cdot \cdot \ 5 \ \cdot \cdot \cdot \cdot \cdot \cdot \ 6\\
& = 6 \ \text{total outcomes}\end{align*}
For selecting a day of the week:
\begin{align*}\text{total outcomes} & = \text{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}\\
& = 7 \ \text{total outcomes}\end{align*}
Favorable outcomes are the specific outcomes you are looking for.
For flipping a coin and having it come up heads:
\begin{align*}\text{favorable outcomes} & = \text{heads}\\ & = 1 \ \text{favorable outcome}\end{align*}
For tossing a number cube and having it come with up an even number:
\begin{align*}\text{favorable outcomes} & = \cdot \ 1 \ \cdot \cdot \ 2 \ \cdot \cdot \cdot \ 3 \ \cdot \cdot \cdot \cdot \ 4 \ \cdot \cdot \cdot \cdot \cdot \ 5 \ \cdot \cdot \cdot \cdot \cdot \cdot \ 6\\ & = 3 \ \text{favorable outcomes}\end{align*}
For randomly choosing a date and have it land on a weekday:
\begin{align*}\text{favorable outcomes} & = \text{Monday, Tuesday, Wednesday, Thursday, Friday}\\ & = 5 \ \text{favorable outcomes}\end{align*}
To write a ratio, we compare the favorable outcome to the total outcomes. Comparing favorable outcomes to possible total outcomes is what we call theoretical probability.
II. Calculate Simple Theoretical Probabilities
Now that you know how to think about theoretical probability, we can look at calculating some simple probabilities. Remember, that we are going to be writing ratios that compare favorable outcomes with total outcomes.
The probability of any event is written as \begin{align*}P (\text{event})\end{align*}. So:
\begin{align*}P(A)\end{align*} is the probability that event \begin{align*}A\end{align*} will occur.
\begin{align*}P (\text{heads})\end{align*} is the probability that heads will turn up on a flipped coin.
\begin{align*}P (5)\end{align*} is the probability that a number cube will turn up as 5
Look at this example.
Example
What is the probability of flipping heads on a coin?
To work through this probability, we are going to be writing a ratio that compares the number of favorable outcomes with the total number of outcomes.
Favorable outcome = 1, since there is one heads on a coin
Total outcomes = 2, since there is the possibility of heads or tails
The answer is 1:2.
Example
For tossing a number cube and having it land an even number:
Our final answer is 1:2.
To find any probability, follow the steps below.
Problem: What is the probability of the arrow landing on a yellow section?
Step 1: Count the number of favorable outcomes.
There are 2 yellow spaces, so
favorable outcomes = 2
Step 2: Count the number of total outcomes.
There are 5 spaces in all, so
total outcomes = 5
Step 3: Write the ratio of favorable outcomes to total outcomes
\begin{align*}P (\text{yellow}) &= \text{favorable outcomes} : \text{total outcomes}\\ &= 2:5\end{align*}
Take a few minutes to write these steps in your notebook.
Now let’s apply these steps to an example.
Example
What is the probability of the arrow landing on a silver or pink section?
Step 1: Count the number of favorable outcomes.
There are 2 silver spaces and 1 pink space, so
favorable outcomes = 3
Step 2: Count the number of total outcomes.
There are 5 spaces in all, so
total outcomes = 5
Step 3: Write the ratio of favorable outcomes to total outcomes
\begin{align*}P (\text{silver or pink}) = \text{favorable outcomes} : \text{total outcomes} = 3:5\end{align*}
Our answer is 3:5.
12A. Lesson Exercises
Write each theoretical probability. Be sure to simplify the ratio when necessary.
- What is the probability of rolling a 1 or a 3 on a number cube?
- If there are four blue marbles and one red marble in a bag, what is the probability of pulling out a red one?
- What is the probability of pulling out a blue one?
Take a few minutes to check your work with a friend.
III. Write Theoretical Probabilities as Fractions, Decimals or Percents
Now that you know about probability and how to calculate a theoretical probability, we can look at the different forms that probabilities can take.
Remember back to our work with ratios in an earlier chapter? You learned that ratios can be written as fractions, decimals and percents.
Example
1:2 can be written as \begin{align*}\frac{1}{2}\end{align*} or as a decimal of .50 or as a percent of 50%
Because probability is written as a ratio, we can also write probabilities as fractions, decimals and/or percents.
Let’s look at doing this with an example.
Example
A bag contains 5 black ping pong balls, 8 white ping pong balls, and 7 yellow ping pong balls. What is the probability of drawing a black ping pong ball from the bag?
First, let’s look at writing a ratio.
\begin{align*}P (\text{black}) &= \frac{\text{favorable outcomes}}{\text{total outcomes}}\\ \text{favorable outcomes} &= 5\\ \text{total outcomes} &= 20\\ P (\text{black}) &= 5:20\\ &= 1:4\end{align*}
Now that we have a ratio, we can easily take this ratio and write it as a fraction. Notice that the first value in the ratio becomes the numerator and the second number becomes the denominator.
1:4 becomes \begin{align*}\frac{1}{4}\end{align*}
Next, we can take this and convert it to a decimal. There are two ways to do this.
The first way is to divide the numerator by the denominator.
\begin{align*}\overset{ \ \ .25}{4 \overline{ ) {1.00 \;}}}\end{align*}
The decimal is .25.
The second way is to write a proportion. We can convert one-fourth to a proportion out of 100.
\begin{align*}\frac{1}{4} &= \frac{\Box}{100}\\ \frac{1}{4} &= \frac{25}{100}\end{align*}
Now we can write it as a decimal.
.25
The proportion shows us how to convert the decimal to a percent easily.
.25 or \begin{align*}\frac{25}{100}=25\%\end{align*}
Now we have written the probability all four possible ways.
Let’s look at another example.
Example
Thinking about the bag with the ping pong balls, what is the probability of choosing a yellow ping pong ball?
\begin{align*}P (\text{yellow}) &= \frac{\text{favorable outcomes}}{\text{total outcomes}}\\ \text{favorable outcomes} &= 7\\ \text{total outcomes} &= 20\\ P (\text{yellow}) &= 7:20\end{align*}
Now let’s write this as a fraction.
\begin{align*}7:20 = \frac{7}{20}\end{align*}
As a decimal:
\begin{align*}\frac{7}{20}=\frac{35}{100}=.35\end{align*}
And as a percent:
\begin{align*}.35 = 35\%\end{align*}
Remember you can also convert a decimal to a percent by moving the decimal point two places to the right and then adding a percent sign.
12B. Lesson Exercises
Write each ratio as a fraction, decimal and percent.
- 1:5
- 2:4
- 4:5
Check your answers with a partner. Then continue with the lesson.
IV. Make Predictions Based on Theoretical Probabilities
A prediction is a reasonable guess about what will happen in the future. Good predictions should be based on facts. For example, you might predict that it’s going to rain today. Your prediction is reasonable if it is based on facts and evidence. For example, you might base your prediction on:
- a reliable weather report you heard
- dark clouds in the sky
- a satellite photo of your area
- a phone call from a nearby friend who lives where it’s raining
The most accurate kinds of predictions are based on probability.
For example, you might predict that a single spin of the spinner above is likely to turn out red. Why? If you assume that all 5 spinner sections are equal in size and equally likely to be landed on, then the probability of spinning red is:
\begin{align*}P (\text{red}) = \frac{favorable \ outcomes}{total \ outcomes}=\frac{3}{5}\end{align*}
Since a 3 out of 5 probability is greater than 50 percent, it is reasonable to predict that the spinner is likely to land on red.
Here are some other reasonable predictions you might make.
Reasonable Prediction | Reason |
---|---|
The spinner is more likely to land on red than blue. | There are 3 red sections and only 1 blue section. |
The spinner is equally likely to land on blue or yellow. | There spinner has an equal number of blue and yellow sections – one each. |
On the average, a ratio of about 4 of 5 spins are likely turn up blue or red. | Four out of 5 sections are blue or red. |
Most of the time the spinner will not land on blue or yellow. | Blue and yellow combined make up fewer than half of the sections. |
Not all predictions are reasonable.
For example, take a look at these unreasonable predictions.
Unreasonable Prediction | Reason |
---|---|
The spinner is more likely to land on blue before it lands on red. | There are fewer blue sections (1) than red sections (3). |
The spinner will never land on blue before it lands on red. | The spinner can land on blue before it lands on red; it just isn’t likely to occur. |
Three out of every 5 spins will always be red. | On the average, 3 of 5 spins will be red, but on any given series of spins anything can happen. |
When making predictions it’s best to keep in mind that probability predicts only what is likely to happen. All events are subject to chance. On any given event, anything can happen. If you flip a coin 4 times, you are most likely to land on heads twice and tails twice.
However, that doesn’t mean that on any given set of 4 flips you might land on heads zero times, or you might land on heads 4 times. Neither event is likely, but both are possible and each will happen from time to time.
So now you might ask – what good is probability for making predictions if you can’t rely on it to be true every time? In fact, the value of probability is very limited in the short term. But over the long term, predictions based on probability are usually highly accurate.
In general, the greater the number of outcomes you have, the closer a prediction based on probability is likely to be.
Let’s look at an example.
Example
To make predictions using data, use the table and follow the steps below.
Pizza shop | Favorite |
---|---|
Pizza Town | 10% |
Hot ‘\begin{align*}n\end{align*}’ Tasty | 25% |
Joe Shmoe’s | 35% |
The Noble Pie | 20% |
Problem: Out of 90 orders, predict how many customers will call Pizza Town for pizza.
You can see that the percent of people who called Pizza Town was 10%. We know that there were 90 orders. To find the number of orders from Pizza Town, we need to find 10% of the 90 orders.
Solution: Find 10 percent of the total.
\begin{align*}10\% \ \text{of} \ 90 &= 0.1 \cdot 90\\ &= 9\end{align*}
So you would predict that 9 out of 90 customers would order from Pizza Town.
12C. Lesson Exercises
Use the data to answer the following questions.
Data for Joe Shmoe’s Pizza is shown in the table.
Pizza type | Favorite | Pizza type | Favorite |
---|---|---|---|
cheese | 35% | sausage | 15% |
pepperoni | 25% | mushroom | 5% |
green peppers | 10% | spinach | 3% |
onion | 5% | hot peppers | 2% |
- Out of 40 orders, predict how many customers will order pepperoni pizza.
- Out of 100 orders, predict how many customers will order spinach pizza.
- Out of 60 orders, predict how many customers will order either sausage or pepperoni pizza.
- Out of 80 orders, predict which will be greater, the number of sausage pizzas or green pepper pizzas.
Check your answers with a friend. Be sure to correct any errors before continuing.
Real–Life Example Completed
The Talent Show
Here is the original problem once again. Reread it and then answer the following questions about probability.
Since the opening of J.S. Middle School, a tradition has been the end of the year talent show. The school opened ten years ago, and within that time there have been 8 talent shows. There were two years when the school was not able to host one because there was flooding or repairs were being done in auditorium.
“I wonder if we are going to have the talent show this year,” Carmen asked at lunch one day.
“I am sure that we are,” Tyler said, biting into his ham sandwich. “After all, there were only two years that the talent show did not happen and that was because of the circumstances.”
“Well, are there any circumstances this year?”
“I don’t think so. The probability is high that is going to happen.”
“What is the probability of the talent show happening?” Carmen asked, taking a sip of milk.
To think about the probability of the talent show happening, we can take the data from the past ten years and create a ratio.
\begin{align*}\text{Probability} = \frac{favorable \ outcomes}{total \ outcomes}\end{align*}
In the past ten years, there have been 8 talent shows. There have been ten possible years to calculate with. These are the total outcomes. Here is our ratio.
8:10
We can write this as a fraction, a decimal and a percent too.
\begin{align*}\frac{8}{10}=.8=80\%\end{align*}
There is an 80% chance that the talent show will happen. These are very good odds-it is very likely to occur.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Probability
- the likelihood that an event will happen.
- Event
- result of an experiment or an activity
- Favorable Outcome
- the outcome that you are looking for
- Total Outcomes
- the total number of possible outcomes
- Ratio
- a comparison of two quantities
- Theoretical Probability
- the ratio that compares the number of favorable outcomes to the number of total outcomes.
- Prediction
- a reasonable guess about a future event.
Technology Integration
Khan Academy, Basic Probability
James Sousa, Introduction to Probability
James Sousa, Determining Probability
Time to Practice
Directions: Answer each question or solve each problem as it connects to probability.
1. For rolling a 4 on the number cube:
(a) List each favorable outcome.
(b) Count the number of favorable outcomes.
(c) Write the total number of outcomes.
2. For rolling a number greater than 2 on the number cube:
(a) List each favorable outcome.
(b) Count the number of favorable outcomes.
(c) Write the total number of outcomes.
3. For rolling a 5 or 6 on a number cube:
(a) List each favorable outcome.
(b) Count the number of favorable outcomes.
(c) Write the total number of outcomes.
4. A box contains 12 slips of paper numbered 1 to 12. For randomly choosing a slip with an even number on it:
(a) List each favorable outcome.
(b) Count the number of favorable outcomes.
(c) Write the total number of outcomes.
5. A box contains 12 slips of paper numbered 1 to 12. For randomly choosing a slip with a number greater than 3:
(a) List each favorable outcome.
(b) Count the number of favorable outcomes.
(c) Write the total number of outcomes.
6. For randomly choosing a marble and having it turn out to be orange:
(a) Count the number of favorable outcomes.
(b) Write the total number of outcomes.
7. For randomly choosing a marble and having it turn out to be large:
(a) Count the number of favorable outcomes.
(b) Write the total number of outcomes.
8. For randomly choosing a marble and having it turn out to be blue:
(a) Count the number of favorable outcomes.
(b) Write the total number of outcomes.
9. For randomly choosing a marble and having it turn out to be small:
(a) Count the number of favorable outcomes.
(b) Write the total number of outcomes.
10. For randomly choosing a marble and having it turn out to be orange and large:
(a) Count the number of favorable outcomes.
(b) Write the total number of outcomes.
11. What is the probability of the spinner landing on 9?
(a) List each favorable outcome.
(b) Count the number of favorable outcomes.
(c) Count the total outcomes.
(d) Write the probability. Simplify, if necessary.
12. What is the probability of the spinner landing on 3 or 4?
(a) List each favorable outcome.
(b) Count the number of favorable outcomes.
(c) Count the total outcomes.
(d) Write the probability. Simplify, if necessary.
13. What is the probability of the spinner landing on blue?
(a) List each favorable outcome.
(b) Count the number of favorable outcomes.
(c) Count the total outcomes.
(d) Write the probability. Simplify, if necessary.
14. What is the probability of the spinner landing on a silver number greater than 4?
(a) List each favorable outcome.
(b) Count the number of favorable outcomes.
(c) Count the total outcomes.
(d) Write the probability. Simplify, if necessary.
Directions: Answer each question and write the probability as a fraction, a decimal and a percent.
15. A clothes dryer contains 12 socks. What is the probability of reaching inside the dryer and pulling out a blue sock?
(a) List each favorable outcome.
(b) Count the number of favorable outcomes.
(c) Write the probability.
16. What is the probability of pulling a red sock out of the dryer?
(a) List each favorable outcome.
(b) Count the number of favorable outcomes.
(c) Write the probability.
17. What is the probability of pulling a blue or white sock out of the dryer?
(a) List each favorable outcome.
(b) Count the number of favorable outcomes.
(c) Write the probability.
18. What is the probability of pulling a blue or red sock out of the dryer?
(a) List each favorable outcome.
(b) Count the number of favorable outcomes.
(c) Write the probability.