12.3: Experimental Probability
Jackson has just been told by his football coach that he will be the starting kicker in this week's football game, meaning the team will be depending on him to kick the field goals. Jackson wants a little reassurance before the game that he will have a successful first starting game. He decides to count the number of successful field goals he kicks. Out of 20 tries, he kicks 16 field goals and misses 4 times. What is the experimental probability that Jackson will kick a field goal during the game?
In this concept, you will learn about experimental probability.
Experimental Probability
Probability is based on simple facts. For example, since there are two sides of a coin, heads and tails, and each side has an equal chance of turning up, it makes sense to say the probability of heads turning up is 50 percent, or 1 out of 2.
However, making sense and being true are often two different things. It may make sense to say the probability of heads is 50 percent, but does a real coin in the real world actually turn up heads 50 percent of the time? To answer that question, you need to learn about empirical probability, also called 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 spinner landing on the red section you would need to conduct an experiment. Suppose you were to spin the spinner 60 times.
Favorable outcomes: red
Total outcomes: 60 spins
\begin{align*}\underline{\text{Experimental probability}}: \ P (\text{red}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}=\frac{\text{number of red}}{\text{total number of spins}}\end{align*}
Now you can use this data to conduct an experiment.
What is the experimental probability of having the arrow of the spinner land in the red section?
Trial | 1 | 2 | 3 | 4 | 5 | 6 | Total |
---|---|---|---|---|---|---|---|
red favorable outcomes | 2 | 3 | 1 | 5 | 2 | 4 | 17 |
total spins total outcomes | 10 | 10 | 10 | 10 | 10 | 10 | 60 |
experimental probability: ratio of favorable outcomes to total outcomes | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | 17:60 |
Solve the problem by spinning the spinner 60 times in 6 trials of 10 spins each as shown in the table above. Then follow the steps below.
First, total up the number of favorable outcomes – the number of times the spinner landed on red. This is shown in red in the final column of the table as 17.
Next, add up the number of total outcomes – the total number of spins. This is shown in red in the final column of the table as 60.
Then, compute the experimental probability – the ratio of favorable outcomes to total outcomes in percent form. The experimental probability, in ratio form, is 17:60. Convert to a percentage:
\begin{align*}\overset{ \ \ .28}{60 \overline{ ) {17.00 \;}}}\end{align*} = 28%
The answer is the experimental probability of favorable outcomes to total outcomes is 28%.
A number cube was rolled in a probability experiment 40 times. The results are shown in the table. Compute the experimental probability of rolling a 5.
Dots on number cube | \begin{align*}\cdot\end{align*} | \begin{align*}\cdot \cdot\end{align*} | \begin{align*}\cdot \cdot \cdot\end{align*} |
\begin{align*}\cdot \cdot\end{align*} \begin{align*}\cdot \cdot\end{align*} |
\begin{align*}\cdot \cdot \cdot\end{align*} \begin{align*}\cdot \cdot\end{align*} |
\begin{align*}\cdot \cdot \cdot\end{align*} \begin{align*}\cdot \cdot \cdot\end{align*} |
Total |
---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | ||
number of times cube landed | \begin{align*}\cancel{||||}\end{align*} | \begin{align*}| | | |\end{align*} | \begin{align*}\cancel{| | | |}\end{align*} | \begin{align*}\cancel{| | | |}\end{align*} | \begin{align*}\cancel{| | | |}\end{align*} | \begin{align*}\cancel{| | | |}\end{align*} | \begin{align*}x\end{align*} |
\begin{align*}|\end{align*} | \begin{align*}||||\end{align*} | \begin{align*}|||\end{align*} | \begin{align*}|\end{align*} | \begin{align*}||\end{align*} | |||
total from tally | 6 | 4 | 9 | 8 | 6 | 7 | 40 |
favorable outcomes | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | 6 | \begin{align*}x\end{align*} | 6 |
experimental probability: ratio of favorable outcomes to total outcomes | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | 6:40 = 3:20 |
First, add up the tallies to get the number of total number of outcomes. This is recorded as 40 in the fifth row of the table.
Next, find the number of favorable outcomes – the number of times the number cube landed on 5. This is recorded as 6 in the sixth row of the table.
Then compute the experimental probability by finding the ratio of favorable outcomes to total outcomes, making sure to simplify if possible: 6:40 = 3:20
Then answer is the experimental probability of rolling a 5 a 3:20 ratio.
Most probability experiments are conducted as a process of testing a hypothesis. A hypothesis is a statement that you want to test to see if it’s true.
Here are some examples of hypotheses.
- A flipped coin comes up heads 50 percent of the time. True or false?
- A number cube will land on three \begin{align*}\frac{1}{6}\end{align*} th of the time. True or false?
- It rains more on the weekend than it does on weekdays. True or false?
- You perform better on math tests when you get a good night’s sleep. True or false?
- A flipped coin will land on heads \begin{align*}\frac{1}{2}\end{align*}, or 50 percent of the time. True or false?
To compute the experimental probability of the hypothesis you would need to conduct an experiment. Suppose you were to flip the coin 50 times. To find out how often it lands on heads, follow the steps below.
First, state your hypothesis including the theoretical probability and make a prediction.
Hypothesis: The coin will land on heads \begin{align*}\frac{1}{2}\end{align*}, or 50 percent of the time.
Use the theoretical probability to make a prediction. Since there are 2 different outcomes, and 1 of those outcomes is heads, it make sense to predict that heads will come up 1 out of 2, or 50 percent of the time.
\begin{align*}P (\text{heads}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{1}{2} \Longrightarrow \text{prediction:heads} = 50\%\end{align*}
Next, conduct an experiment and collect data. You can use a tally table like the one shown below. Fill in your predicted values and tally as you conduct the experiment.
Trial | 1 | 2 | 3 | 4 | 5 | Prediction | Total |
---|---|---|---|---|---|---|---|
tally | \begin{align*}| | | |\end{align*} | \begin{align*}\cancel{| | | | |}\end{align*} | \begin{align*}| | |\end{align*} | \begin{align*}\cancel{| | | | |}\end{align*} | \begin{align*}\cancel{| | | | |}\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} |
\begin{align*}|\end{align*} | |||||||
heads | 4 | 5 | 3 | 6 | 5 | 25 | 23 |
total flips |
10 | 10 | 10 | 10 | 10 | 50 | 50 |
Then, analyze your data and see how well it agrees with your prediction. You can see that 23 out of 50 flips came up heads. The data does not perfectly agree with the prediction, but 23 is close to 25. So the results are fairly close to the prediction. In general, the more total outcomes included in the experiment, the more likely the experimental probability will agree with the predicted probability.
The answer is the prediction is 50% and the experimental probability is
\begin{align*} \frac{23}{50} = 46 \%\end{align*} .
Examples
Example 1
Earlier, you were given a problem about Jackson's experiment to predict the probability of him kicking a field goal during his football game.
He kicked the ball 20 times. He kicked 16 field goals and missed 4 times. What is the experimental probability that Jackson will kick a field goal during the game?
First, determine the number of successful outcomes:
Jackson succeeded in kicking 16 field goals. There are 16 successful outcomes.
Next, detemine the total number of attempts (trials attempted):
Jackson attempted to kick a field goal 20 times. There are 20 trials attempted.
Then, create a ratio from the data and convert to a percent:
\begin{align*}\text{Probability} &= \frac{successful \ outcomes}{trials \ attempted}\\ 16:20 &= \frac{16}{20}\\ \frac{16}{20} &= 0.8 = 80\ percent \end{align*}The answer is Jackson's experimental probability of kicking a field goal during the football game is 80%.
Example 2
Use the table to compute the experimental probability of the arrow in the spinner landing on yellow and record your answer as a percentage.
Trial | 1 | 2 | 3 | 4 | 5 | 6 | Total |
---|---|---|---|---|---|---|---|
number of times arrow landed on yellow | 4 | 2 | 3 | 1 | 2 | 5 | |
total number of spins | 10 | 10 | 10 | 10 | 10 | 10 | |
experimental probability | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} |
First, calculate the total attempts:
There were 6 x 10 total attempts to have the spinner land on yellow.
Next, calculate the number of attempts that landed on yellow:
If you add up each success out of 10 we have: 4 + 2 + 3 + 1 + 2 + 5 = 17
Then, record the results as a ratio and convert to a percent.
\begin{align*}17:60= \frac{17}{60} = .2833 = 28\%\end{align*}
The answer is the experimental probability of the spinner landing on yellow is 28% of the time.
Use the below information the answer the questions that follow:
Jennifer attempted to juggle three pins for five minutes. She made five attempts and dropped a club twice.
Example 3
Write a probability that shows successful attempts.
First, determine the number of successful attempts (favorable outcomes):
The number of successful attempts is 3.
Next, determine the total number of attempts (total outomes):
The total number of attempts is 5.
Then, create a ratio out of the data: 3:5
The answer is the probability of successful attempts is 3:5.
Example 4
Write a probability that shows drops.
First, determine the number of successful attempts (favorable outcomes):
If the number of successful attempts is 3 out of a total of 5 total attempts, then there were 2 drops.
Next, determine the total number of attempts (total outomes):
The total number of attempts is 5.
Then, create a ratio out of the data: 2:5
The answer is the probability of drops is 2:5.
Example 5
What is the percent of success?
First, convert the ratio into a fraction: 3:5 = 3/5
Next, divide the numerator by the denominator:
\begin{align*}\overset{ \ \ .60}{5 \overline{ ) {3.00 \;}}}\end{align*}
Then, convert the decimal value into a percentage: 0.60 = 60%
The answer is the probability of successful attempts is 60%.
Review
Use what you have learned about experimental probability to complete each problem.
Use the table to compute the experimental probability of the arrow in the spinner landing on yellow.
Trial | 1 | 2 | 3 | 4 | 5 | 6 | Total |
---|---|---|---|---|---|---|---|
number of times arrow landed on yellow | 4 | 2 | 3 | 1 | 2 | 5 | |
total number of spins | 10 | 10 | 10 | 10 | 10 | 10 | |
experimental probability | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} |
- 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?
Use the table to compute the experimental probability of the arrow in the spinner above landing on blue or green.
Trial | 1 | 2 | 3 | 4 | 5 | Total |
---|---|---|---|---|---|---|
number of times arrow landed on blue or green | 4 | 5 | 6 | 5 | 4 | |
total spins | 10 | 10 | 10 | 10 | 10 | |
experimental probability | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} |
- 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?
Use the table to compute the experimental probability of the arrow landing on any color but blue.
Trial | 1 | 2 | 3 | 4 | 5 | Total |
---|---|---|---|---|---|---|
number of times arrow landed on any color but blue | 15 | 17 | 14 | 16 | 16 | |
total spins | 20 | 20 | 20 | 20 | 20 | |
experimental probability |
- 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?
A number cube was rolled in a probability experiment 40 times. The results are shown in the table. Compute the experimental probability of rolling a 2.
Dots on number cube | \begin{align*}\cdot\end{align*} | \begin{align*}\cdot \cdot\end{align*} | \begin{align*}\cdot \cdot \cdot\end{align*} |
\begin{align*}\cdot \cdot\end{align*} \begin{align*}\cdot \cdot\end{align*} |
\begin{align*}\cdot \cdot \cdot\end{align*} \begin{align*}\cdot \cdot\end{align*} |
\begin{align*}\cdot \cdot \cdot\end{align*} \begin{align*}\cdot \cdot \cdot\end{align*} |
Total |
---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | ||
number of times cube landed | \begin{align*}\cancel{| | | |}\end{align*} | \begin{align*}\cancel{| | | |}\end{align*} | \begin{align*}\cancel{| | | |}\end{align*} | \begin{align*}\cancel{| | | |}\end{align*} | \begin{align*}\cancel{| | | |}\end{align*} | \begin{align*}| | | |\end{align*} | \begin{align*}x\end{align*} |
\begin{align*}||||\end{align*} | \begin{align*}|||\end{align*} | \begin{align*}|\end{align*} | \begin{align*}||\end{align*} | \begin{align*}|\end{align*} | |||
total from tally | 9 | 8 | 6 | 7 | 6 | 4 | |
favorable outcomes | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | ||
experimental probability | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} |
- 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 rolling a 2?
A number cube was rolled in a probability experiment 50 times. The results are shown in the table. Compute the experimental probability of rolling a 3 or a 4.
Dots on number cube | \begin{align*}\cdot\end{align*} | \begin{align*}\cdot \cdot\end{align*} | \begin{align*}\cdot \cdot \cdot\end{align*} |
\begin{align*}\cdot \cdot\end{align*} \begin{align*}\cdot \cdot\end{align*} |
\begin{align*}\cdot \cdot \cdot\end{align*} \begin{align*}\cdot \cdot\end{align*} |
\begin{align*}\cdot \cdot \cdot\end{align*} \begin{align*}\cdot \cdot \cdot\end{align*} |
Total |
---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | ||
number of times cube landed | \begin{align*}\cancel{| | | |}\end{align*} | \begin{align*}\cancel{| | | |}\end{align*} | \begin{align*}\cancel{| | | |}\end{align*} | \begin{align*}\cancel{| | | |}\end{align*} | \begin{align*}\cancel{| | | |}\end{align*} | \begin{align*}\cancel{| | | |}\end{align*} | \begin{align*}x\end{align*} |
\begin{align*}\cancel{||||}\end{align*} | \begin{align*}|\end{align*} | \begin{align*}||||\end{align*} | \begin{align*}||\end{align*} | \begin{align*}\cancel{||||}\end{align*} | \begin{align*}||\end{align*} | ||
\begin{align*}|\end{align*} | |||||||
total from tally | 10 | 6 | 9 | 7 | 11 | 7 | |
favorable outcomes | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | 9 | 7 | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | |
experimental probability | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} | \begin{align*}x\end{align*} |
- 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 rolling a 2?
Review (Answers)
To see the Review answers, open this PDF file and look for section 12.3.
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