11.4: Theoretical and Experimental Probability
Introduction
Weekend Woes
“I don’t want to work on the weekend,” Carey said to Telly at lunch one day.
“But that was part of the deal. We both have to work one day out of the weekend,” Telly said.
“Well, which day do you want?” Carey asked.
“I don’t know. I haven’t really thought about it,” Telly said. “But we could make it really random.”
“How?” Carey asked.
Telly took two pieces of paper and wrote Saturday on one and Sunday on the other.
“Now we can figure out the probability of you getting Saturday or Sunday,” she said.
We can stop there. This lesson is all about probability. Telly’s experiment is an example of experimental probability. Let’s talk more about this at the end of the lesson.
What You Will Learn
In this lesson, you will learn how to demonstrate the following skills.
- Recognize the theoretical probability of an event as the ratio of favorable outcomes to possible outcomes.
- Recognize the experimental probability of an event as the ratio of successful outcomes to trials attempted.
- Write and compare probabilities as fractions, decimals and percents.
- Make and compare predictions based on theoretical and experimental probabilities, justifying the use of either.
Teaching Time
I. Recognize the Theoretical Probability of an Event as the Ratio of Favorable Outcomes to Possible Outcomes
In this chapter we have explored different outcomes for events, but not probability itself. Probability is defined as a mathematical way of calculating how likely an event is to occur. The probability of an event occurring is defined as the ratio of favorable outcomes to the number of possible equally likely 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*}
Theoretical probability is probability that is based on an ideal situation. For example, since a flipped coin has two sides and each side is equally likely to land up, the theoretical probability of landing heads (or tails) is exactly 1 out of 2. Whether or not the coin actually lands on heads (or tails) 1 out of every 2 flips in the real world does not affect theoretical probability. The theoretical probability of an event remains the same no matter how events turn out in the real world.
Example
Find the probability of tossing a number cube and having it come up “4”.
Step 1: Find the total number of outcomes
\begin{align*}\text{Total outcomes} &= \bullet \ 1 \ \bullet \bullet \ 2 \ \bullet \bullet \bullet \ 3 \ \bullet \bullet \bullet \bullet \ 4 \ \bullet \bullet \bullet \bullet \bullet \ 5 \ \bullet \bullet \bullet \bullet \bullet \bullet \ 6 \\
&= 6 \ \text{total outcomes}\end{align*}
Step 2: Find the number of favorable outcomes.
\begin{align*}\text{favorable outcomes} &= \bullet \ 1 \ \bullet \bullet \ 2 \ \bullet \bullet \bullet \ 3 \ \ {\color{red}\bullet \bullet \bullet \bullet \ 4} \ \bullet \bullet \bullet \bullet \bullet \ 5 \ \bullet \bullet \bullet \bullet \bullet \bullet \ 6 \\
&= 1 \ \text{favorable outcome}\end{align*}
Step 3: Find the ratio of favorable outcomes to total outcomes.
\begin{align*}\text{Favorable}:\text{Total}= 1:6\end{align*}
While theoretical probability is based on the ideal, we can also figure out experimental probability.
II. Recognize the Experimental Probability of an Event as the Ratio of Successful Outcomes to Trials Attempted
Theoretical probability is based on an ideal situation. Since a flipped coin seems equally likely to land up or down, the theoretical probability of landing heads (or tails) is 1 out of 2. Whether or not the coin actually lands on heads (or tails) 1 out of every 2 flips in the real world is something you must determine with experimental probability.
Experimental probability is probability based on doing actual experiments – flipping coins, spinning spinners, picking ping pong balls out of a jar, and so on. To compute the experimental probability of the number cube landing on 3 you would need to conduct an experiment. Suppose you were to toss the number cube 60 times.
Favorable outcomes:
Total outcomes: 60 tosses
Experimental probability:
\begin{align*}P(3) =\frac{favorable \ outcomes}{total \ outcomes}=\frac{Number \ of \ 3's}{Total \ Number \ of \ tosses}\end{align*}
Write this comparison down in your notebooks.
Example
What is the experimental probability of having the number cube land on 3?
trial | 1 | 2 | 3 | 4 | 5 | 6 | Total |
---|---|---|---|---|---|---|---|
raw data:3s | \begin{align*}{|}\end{align*} | \begin{align*}{|||}\end{align*} | \begin{align*}{|}\end{align*} | \begin{align*}{||}\end{align*} | \begin{align*}{||}\end{align*} | ||
favorable outcomes:3s | 1 | 3 | 0 | 1 | 2 | 2 | 9 |
total tosses total outcomes | 10 | 10 | 10 | 10 | 10 | 10 | 60 |
experimental probability: favorable outcomes to total outcomes | x | x | x | x | x | x | \begin{align*}9:60=3:20\end{align*} |
The data from the experiment shows that 3 turned up on the number cube 9 out of 60 times. Simplified, this ratio becomes:
\begin{align*}\text{Favorable outcomes}:\text{total outcomes}= 3:20\end{align*}
You can see that it is only possible to calculate the experimental probability when you are actually doing experiments and counting results.
Example
A spinner was spun in a probability experiment 48 times. The results are shown in the table. Compute the experimental probability of the spinner landing on yellow.
color | red | green | yellow | Total spins |
---|---|---|---|---|
raw data | \begin{align*}\cancel{||||} \ \cancel{||||} \ \cancel{||||} \ {|}\end{align*} | \begin{align*}\cancel{||||} \ \cancel{||||} \ {||||}\end{align*} | \begin{align*}\cancel{||||} \ \cancel{||||} \ \cancel{||||} \ {|||}\end{align*} | 48 |
total from tally | 16 | 14 | 18 | 48 |
favorable outcomes:yellow | x | x | 18 | x |
experimental probability: | x | x | \begin{align*}\colorbox{yellow}{\color{red}18:48}\end{align*} | x |
The data from the experiment above shows that the arrow landed on yellow 18 out of 48 times. Simplified, this ratio becomes:
\begin{align*}\text{Favorable outcomes} : \text{total outcomes} &= 18:48\\ &= 3:8\end{align*}
Notice that the ratio 18 out of 48 was simplified to 3 out of 8. We can simplify probabilities because they are written in ratio form.
III. Write and Compare Probabilities as Fractions, Decimals and Percents
You’ve seen how to compute probabilities in terms of ratios. Since any ratio can be turned into a fraction, decimal, or percent, you can also turn any probability into a fraction, decimal, or percent.
For example, when you toss a number cube, the probability of rolling a “3” is:
\begin{align*}P (3) &=\text{favorable outcomes} : \text{total outcomes}\\ P (3) &= 1 : 6\end{align*}
You can write the same probability as a fraction simply by rewriting the two numbers in the ratio as the numerator and denominator of a fraction.
\begin{align*}P(3)=\frac{1}{6}\end{align*}
This is the answer in fraction form.
How can we turn this fraction into a decimal?
You can turn a fraction into a decimal by dividing the numerator by the denominator.
\begin{align*}\frac{{\color{blue}1}}{{\color{red}6}}=\overset{ \ \ 0.167}{{\color{red}6} \overline{ ) {{\color{blue}1}.000 \;}}}\end{align*}
How can we turn the decimal into a percent?
We can turn a decimal into a percent by multiply the decimal by 100 since a percent is out of 100. Then we can move the decimal point two decimal places to show the percent.
\begin{align*}0.167 = 0.167 \times 100 = 16.7\%\end{align*}
To summarize, the probability of rolling a 3 with a number cube is 1 out of 6, or:
ratio | fraction | decimal | percent |
---|---|---|---|
1:6 | \begin{align*}\frac{1}{6}\end{align*} | 0.167 | 16.7% |
Please write these examples down in your notebook.
IV. Make and Compare Predictions Based on Theoretical and Experimental Probabilities, Justifying the Use of Either
A prediction is a reasonable guess about what will happen in the future. Good predictions should be based on facts and probability. There are two main types of predictions.
Type 1: Predictions based on theoretical probability: These are the most reliable types of predictions, based on physical relationships that are easy to see and measure and that do not change over time. They include such things as:
- coin flips
- spinners
- number cubes
Type 2: Predictions based on data and experimental probability: These predictions are often reliable, but subject to change depending on the situation. They include such activities as:
- batting averages, shooting percentages, and similar data from sports
- predicting the weather
- sales figures from such things as movies, TV shows, products
- polls and surveys that measure opinion
- historical data that measures past events
The difference between the two types of prediction is best illustrated by the following examples.
- Type 1: Coin flip prediction: of 100 flips, 50 are predicted to turn up heads
- Type 2: Weather prediction: a 50 percent chance of rain tomorrow
Note that both predictions are about the future – and you never know what might happen in the future. Though it is HIGHLY UNLIKELY to occur, a coin could land on heads 10, or 20, or even 50 times in a row. Similarly, a weather data might predict a zero percent chance of rain. But on a given day, the unexpected could happen. The winds could change clouds could unexpectedly move out or move in. That’s why predictions based on experimental probability are always less reliable than those based on theoretical probability.
In general, the greater the number of outcomes you have, the closer a prediction based on probability is likely to be.
Write this statement down in your notebooks.
One hundred coin flips should turn out to be close to 50 percent heads; 1000 flips should be even closer to 50 percent; 10,000 flips should be closer yet. The same thing goes with weather forecasts. Over a year, forecasts are a lot more likely to be accurate than over a single day or a single week.
To make predictions, use the following formulas.
\begin{align*}\mathbf{Type \ 1 \ Prediction}= \text{theoretical probability} \cdot \text{number of trials}\end{align*}
Or, for type 2 situations in which you do not have theoretical probability:
\begin{align*}\mathbf{Type \ 2 \ Prediction}= \text{experimental probability} \cdot \text{number of trials}\end{align*}
Problem: Out of 150 number cube rolls, predict how many will turn up greater than 4.
Step 1: Find the theoretical or experimental probability.
\begin{align*}P(> 4) =\frac{favorable \ outcomes}{total \ outcomes}=\frac{2}{6}=\frac{1}{3}\end{align*}
Step 2: Multiply the theoretical probability by the number of total trials
\begin{align*}\text{Prediction} &=\text{theoretical probability} \cdot \text{number of trials}\\ &=\frac{1}{3} \cdot 150 \\ &= 50\end{align*}
Based on these numbers, you would predict that 50 out of 150 rolls would land on numbers greater than 4.
Now let’s apply what you have learned about predictions from the problem in the introduction.
Real-Life Example Completed
Weekend Woes
Here is the original problem from the introduction. Think about how this is an example of experimental probability and then figure out the probability of Carey working Saturday in fraction, decimal and percentage form.
“I don’t want to work on the weekend,” Carey said to Telly at lunch one day.
“But that was part of the deal. We both have to work one day out of the weekend,” Telly said.
“Well, which day do you want?” Carey asked.
“I don’t know. I haven’t really thought about it,” Telly said. “But we could make it really random.”
“How?” Carey asked.
Telly took two pieces of paper and wrote Saturday on one and Sunday on the other.
“Now we can figure out the probability of you getting Saturday or Sunday,” she said.
Remember, there are three parts to your answer.
Solution to Real – Life Example
Carey has a chance of working Saturday or Sunday. There are two possible outcomes. She has a one out of 2 chance of working on Saturday and a one out of two chance of working on Sunday.
\begin{align*}\frac{1}{2}\end{align*}
.50
50% chance or probability for each outcome.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Probability
- a mathematical way of calculating how likely an event is to occur.
- Favorable Outcome
- the outcome that you are looking for
- Total Outcomes
- all of the outcomes both favorable and unfavorable.
- Theoretical Probability
- probability based on an ideal situation relating favorable to total outcomes
- Experimental Probability
- probability based on doing actual experiments.
- Prediction
- a reasonable guess based on probability
Time to Practice
Directions: Solve each problem.
A spinner has five sections: purple, yellow, green, blue and red.
- Find the probability for the arrow landing on blue on the spinner:
- List each favorable outcome.
- Count the number of favorable outcomes.
- Write the total number of outcomes.
- Write the probability.
- Find the probability for the arrow landing on red or green on the spinner:
- List each favorable outcome.
- Count the number of favorable outcomes.
- Write the total number of outcomes.
- Write the probability.
- Find the probability for the arrow NOT landing on yellow on the spinner:
- List each favorable outcome.
- Count the number of favorable outcomes.
- Write the total number of outcomes.
- Write the probability.
- Find the probability for rolling 6 on the number cube:
- List each favorable outcome.
- Count the number of favorable outcomes.
- Write the total number of outcomes.
- Write the probability.
- Find the probability for rolling greater than 2 on the number cube:
- List each favorable outcome.
- Count the number of favorable outcomes.
- Write the total number of outcomes.
- Write the probability.
- Find the probability for rolling less than 4 on the number cube:
- List each favorable outcome.
- Count the number of favorable outcomes.
- Write the total number of outcomes.
- Write the probability.
- Find the probability for rolling 1 or 6 on the number cube:
- List each favorable outcome.
- Count the number of favorable outcomes.
- Write the total number of outcomes.
- Write the probability.
- A box contains 12 slips of paper numbered 1 to 12. Find the probability for randomly choosing a slip with a number less than 4 on it:
- List each favorable outcome.
- Count the number of favorable outcomes.
- Write the total number of outcomes.
- Write the probability.
Directions: Use the table to answer the questions. Express all ratios in simplest form.
Use the table to compute the experimental probability of flipping a coin and having it land on heads.
trial | 1 | 2 | 3 | 4 | 5 | 6 | Total |
---|---|---|---|---|---|---|---|
raw data(heads) | \begin{align*}\cancel{||||}\end{align*} | \begin{align*}\cancel{||||} \ {|}\end{align*} | \begin{align*}\cancel{||||} \ {|}\end{align*} | \begin{align*}{|||}\end{align*} | \begin{align*}\cancel{||||} \ {|}\end{align*} | \begin{align*}\cancel{||||}\end{align*} | |
number of heads | 5 | 6 | 6 | 3 | 6 | 5 | 31 |
total number of flips | 10 | 10 | 10 | 10 | 10 | 10 | 60 |
experimental probability | x | x | x | x | x | x | 31:60 |
- How many favorable outcomes were there in the experiment?
- How many total outcomes were there in the experiment?
- What was the experimental probability of the coin landing on heads? 31:60
Use the table to compute the experimental probability of a number cube landing on 6.
trial | 1 | 2 | 3 | 4 | 5 | Total |
---|---|---|---|---|---|---|
raw data | \begin{align*}{||||}\end{align*} | \begin{align*}{|}\end{align*} | \begin{align*}{|}\end{align*} | \begin{align*}{||}\end{align*} | \begin{align*}{|}\end{align*} | x |
number of 6's | 4 | 1 | 1 | 2 | 1 | 9 |
total tosses | 10 | 10 | 10 | 10 | 10 | 50 |
experimental probability | x | x | x | x | x | 9:50 |
- How many favorable outcomes were there in the experiment?
- How many total outcomes were there in the experiment?
- What is the experimental probability of the arrow landing on yellow?
Directions: Use what you have learned about probability, fractions, decimals and percents to answer each question.
A bag has 6 red marbles, 5 blue marbles, 7 green marbles, 2 white marbles, and 5 yellow marbles. Find the probability of randomly picking out one of the following.
- What is the probability in fraction form of choosing red marble?
- What is the probability in decimal form of choosing green marble?
- What is the probability in percent form of choosing a green or blue marble?
- Which 3 marbles together have a 60 percent chance of being chosen?
- Which 2 marbles together have a 40 percent chance of being chosen?
- Which 3 marbles together have a 0.72 chance of being chosen?
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