12.2: Measurement of Probability
Remember the talent show? Well, take a look.
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.
\begin{align*}8:10\end{align*}
This ratio is what the students came to at the end of the last Concept. Here, the probability is written as a ratio. We can also write probabilities as fractions, decimals and percents.
Do you know how to write this one as a fraction, decimal and percent?
Use this Concept to learn how to write ratios as fractions, decimals and percents.
Guidance
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.
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.
Here is a situation with probabilities.
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 one.
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 for hundredths and then adding a percent sign.
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 pair 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.
Now it's your turn to practice. Write each ratio as a fraction, decimal and percent.
Example A
\begin{align*}1:5\end{align*}
Solution:\begin{align*} \frac{1}{5}, .2 , 20%\end{align*}
Example B
\begin{align*}2:4\end{align*}
Solution:\begin{align*} \frac{1}{2}, .5, 50%\end{align*}
Example C
\begin{align*}4:5\end{align*}
Solution:\begin{align*} \frac{4}{5}, .8, 80%\end{align*}
Here is the original problem once again.
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.
\begin{align*}8:10\end{align*}
This ratio is what the students came to at the end of the last Concept. Here, the probability is written as a ratio. We can also write probabilities as fractions, decimals and percents.
Do you know how to write this one as a fraction, decimal and percent?
Now we want to write the ratio as a fraction, a decimal and a percent. Here is the ratio once again.
\begin{align*}8:10\end{align*}
We can convert this to a fraction and simplify it.
\begin{align*}\frac{8}{10} = \frac{4}{5}\end{align*}
Now we can convert eight to ten to a decimal.
\begin{align*}8:10 = .8\end{align*}
Finally, we convert it to a percent by moving the decimal point two places to the right and adding a % sign.
\begin{align*}.8 = 80%\end{align*}
This is our final answer.
Vocabulary
Here are the vocabulary words in this Concept.
- 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 Outcome
- the total number of possible outcomes
Guided Practice
Here is one for you to try on your own.
Make predictions using data and the table. Answer each question.
Pizza shop | Favorite |
---|---|
Pizza Town | 10% |
Hot ‘\begin{align*}n\end{align*}’ Tasty | 25% |
Joe Shmoe’s | 35% |
The Noble Pie | 20% |
Out of 90 orders, predict how many customers will call Pizza Town for pizza? Then find 10% of the total.
Answer
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.
\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.
Video Review
Here is a video for review.
- This is a James Sousa video on calculating probability.
Practice
Directions: Use what you have learned about theoretical probability to answer each question.
What is the probability of the spinner landing on 9?
1. List each favorable outcome.
2. Count the number of favorable outcomes.
3. Count the total outcomes.
4. Write the probability. Simplify, if necessary.
What is the probability of the spinner landing on 3 or 4?
5. List each favorable outcome.
6. Count the number of favorable outcomes.
7. Count the total outcomes.
8. Write the probability. Simplify, if necessary.
What is the probability of the spinner landing on blue?
9. List each favorable outcome.
10. Count the number of favorable outcomes.
11. Count the total outcomes.
12. Write the probability. Simplify, if necessary.
What is the probability of the spinner landing on a silver number greater than 4?
13. List each favorable outcome.
14. Count the number of favorable outcomes.
15. Count the total outcomes.
16. Write the probability. Simplify, if necessary.
Directions: Answer each question and write the probability as a fraction, a decimal and a percent.
17. A clothes dryer contains 12 socks. What is the probability of reaching inside the dryer and pulling out a blue sock?
18. What is the probability of pulling a red sock out of the dryer?
19. What is the probability of pulling a blue or white sock out of the dryer?
20. What is the probability of pulling a blue or red sock out of the dryer?