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# Empirical Probability

## Identify favorable and total outcomes

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Empirical Probability

Have you ever tried juggling?

Tyler loves juggling. He has been juggling for a long time and loves working on his juggling. He can juggle three balls very easily and can also juggle rings and pins. He very rarely drops anything or makes a mistake.

Tyler has decided to enter his juggling into the Talent show. After practicing for a long time, he has decided to work on juggling four balls instead of three.

“I know this will help me to be a winner,” he tells his Mom.

“It is very impressive, Ty,” She says. “Are you able to do it without dropping any?”

“Well, I’m not sure. Maybe I should do an experiment and see how I am doing.”

Tyler asks his older sister Liz to help him calculate his juggling experiment. Tyler figures that he can attempt to juggle for three minutes without dropping any balls. He figures twenty attempts will give him a good idea of how he will do in the show.

Liz helps Tyler keep track.

She notes that after 20 attempts, that Tyler dropped a ball three times. The other 17 times he was able to juggle for three minutes without dropping anything.

Liz writes this down.

20 attempts

17 successes

3 drops

“How did I do? What is the probability that I will drop a ball?” Tyler asks Liz.

While Tyler and Liz figure out the experimental probability of his dropping a ball, you can take some time to learn about experimental probability in this Concept.

### Guidance

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 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 we 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.

Step 1: 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.

Step 2: 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.

Step 3: Compute the experimental probability – the ratio of favorable outcomes to total outcomes in percent form. The experimental probability, in ratio form, is 17 to 60.

The answer is 17:60.

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

Step 1: Add up the tallies to get the number of total number of outcomes. This is shown in red as 40 in the third row of the table.

Step 2: Find the number of favorable outcomes – the number of times the number cube landed on 5. This is shown in red and highlighted.

Step 3: Compute the experimental probability by finding the ratio of favorable outcomes to total outcomes. The experimental probability of rolling a 5 is 6 to 40. In simplifed form, this is 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?

Clearly, some hypotheses are easier to test than others. It’s much easier to flip coins or roll number cubes than it is to measure weekend rain data or see how sleep affects math test results. To see how experimental probability is measured, consider the hypothesis below.

• 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.

Step 1: State your hypothesis.

The coin will land on heads \begin{align*}\frac{1}{2}\end{align*}, or 50 percent of the time.

Step 1: Compute the theoretical probability

Find the probability of flipping a coin and having it turn up heads. To do this, you need to identify:

• favorable outcomes – the number of times the coin is likely to land on heads
• total outcomes – the total number of flips

Step 2: Make a prediction.

Use 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. (If you need practice in changing ratios to decimals and percents, see 12.1.3.)

\begin{align*}P (\text{heads}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{1}{2} \Longrightarrow \text{prediction:heads} = 50\%\end{align*}

Step 3: Conduct an experiment and collect data. You can use a tally table like the one shown below. Fill in your predicted values first. Then 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*}x\end{align*} \begin{align*}x\end{align*}
total flips 50

Here is what your completed table might look like.

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 10 10 10 10 10 50 50
flips

Step 4: Analyze your data and see how well it agrees with your prediction.

You can see that 23 out of 50 flips came up heads. Does your data agree with your prediction? Not perfectly, but 23 is close to 25, so your results are fairly close to your prediction.

To see how close you are, compare the experimental probability with your prediction in percent form.

\begin{align*}\text{Prediction}: 50 \% \qquad \quad \text{Experimental probability}: \frac{23}{50} = 46 \%\end{align*}

You can see that your experimental probability of 46% agrees fairly well with your predicted probability of 50 percent. In general, the more total outcomes you include in your experiment, the more likely your experimental probability is to agree with your predicted probability.

Now it's time for you to practice writing an experimental probability for each situation.

Jennifer attempted to juggle three pins for five minutes. She made five attempts and dropped a club twice.

#### Example A

Write a probability that shows attempts to successes.

Solution: \begin{align*}\frac{3}{5}\end{align*}

#### Example B

Write a probability that shows drops out of successes.

Solution: \begin{align*}\frac{2}{5}\end{align*}

#### Example C

What is the percent of success?

Solution: \begin{align*}\frac{3}{5} = 60%\end{align*}

Here is the original problem once again.

Tyler loves juggling. He has been juggling for a long time and loves working on his juggling. He can juggle three balls very easily and can also juggle rings and pins. He very rarely drops anything or makes a mistake.

Tyler has decided to enter his juggling into the Talent show. After practicing for a long time, he has decided to work on juggling four balls instead of three.

“I know this will help me to be a winner,” he tells his Mom.

“It is very impressive, Ty,” She says. “Are you able to do it without dropping any?”

“Well, I’m not sure. Maybe I should do an experiment and see how I am doing.”

Tyler asks his older sister Liz to help him calculate his juggling experiment. Tyler figures that he can attempt to juggle for three minutes without dropping any balls. He figures twenty attempts will give him a good idea of how he will do in the show.

Liz helps Tyler keep track.

She notes that after 20 attempts, that Tyler dropped a ball three times. The other 17 times he was able to juggle for three minutes without dropping anything.

Liz writes this down.

20 attempts

17 successes

3 drops

“How did I do? What is the probability that I will drop a ball?” Tyler asks.

Tyler and Liz begin to calculate the experimental probability of the event happening. The event is Tyler not dropping a ball.

Out of 20 attempts, there were 17 attempts where Tyler did not drop a ball.

We can write this ratio to represent the data.

\begin{align*}\text{Probability} &= \frac{successful \ outcomes}{trials \ attempted}\\ 17:20 &= \frac{17}{20}\end{align*}

To get a better idea of the likelihood of this event, we can rewrite the ratio as a percent out of 100.

\begin{align*}\frac{17}{20}=\frac{85}{100}=85\%\end{align*}

Tyler has an 85% chance of juggling for three minutes without dropping a ball.

### Vocabulary

Experimental Probability
probability found by conducting an experiment.
Hypothesis
an educated or reasonable guess based on an idea that one would like to figure out.
Prediction
a statement one makes about the likelihood of a future event.

### Guided Practice

Here is one for you to try on your own.

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*}

There were 6 x 10 total attempts to have the spinner land on yellow.

Each of those attempts had different results.

If we add up each success out of 10 we have: 4 + 2 + 3 + 1 + 2 + 5 = 17

The total attempts was 60.

\begin{align*}\frac{17}{60}\end{align*}

This is our ratio, now we can write it as a percent.

\begin{align*}\frac{17}{60} = .2833 = 28%\end{align*}

The answer is 28%.

### Practice

Directions: 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*}

1. How many favorable outcomes were there in the experiment?

2. How many total outcomes were there in the experiment?

3. 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*}

4. How many favorable outcomes were there in the experiment?

5. How many total outcomes were there in the experiment?

6. 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

7. How many favorable outcomes were there in the experiment?

8. How many total outcomes were there in the experiment?

9. 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*}

10. How many favorable outcomes were there in the experiment?

11. How many total outcomes were there in the experiment?

12. 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*}

13. How many favorable outcomes were there in the experiment?

14. How many total outcomes were there in the experiment?

15. What is the experimental probability of the rolling a 2?

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### Vocabulary Language: English

experimental probability

Experimental (empirical) probability is the actual probability of an event resulting from an experiment.

Hypothesis

A hypothesis is an educated or reasonable guess based on an idea that one would like to figure out.

Prediction

A prediction is a statement one makes about the likelihood of a future event.

Simulation

A way of collecting data using objects such as spinners, coins or cards.