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12.2: Experimental Probability

Created by: CK-12

Introduction

The Juggler

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, 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.

What You Will Learn

By the end of this lesson, you will understand how to demonstrate the following skills.

  • Recognize the experimental probability of events as the ratio of successful outcomes to trials attempted.
  • Calculate simple experimental probability.
  • Write experimental probability as fractions, decimals and percents.
  • Make predictions based on experimental probability given experimental survey or historical data.

Teaching Time

I. Recognize the Experimental Probability of Events as the Ratio of Successful Outcomes to Trials Attempted

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

\underline{\text{Experimental probability}}: \ P (\text{red}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}=\frac{\text{number of red}}{\text{total number of spins}}

Now we can use this data to conduct an experiment. Look at this example.

Example

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 x x x x x x 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 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 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.

Example

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 \cdot \cdot \cdot \cdot \cdot \cdot

\cdot \cdot

\cdot \cdot

\cdot \cdot \cdot

\cdot \cdot

\cdot \cdot \cdot

\cdot \cdot \cdot

Total
1 2 3 4 5 6
number of times cube landed \cancel{||||} | | | | \cancel{| | | |} \cancel{| | | |} \cancel{| | | |} \cancel{| | | |} x
| |||| ||| | ||
total from tally 6 4 9 8 6 7 40
favorable outcomes x x x x 6 x 6
experimental probability: ratio of favorable outcomes to total outcomes x x x x x x 6:40 = 3:20

Step 1: Add up the tallies to get the number of total number of outcomes. This is shown 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.

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.

II. Calculate Simple Experimental Probabilities

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 \frac{1}{6} 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 \frac{1}{2}, 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 \frac{1}{2}, 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 makes 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.)

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

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 | | | | \cancel{| | | | |} | | | x x
heads 25
total flips 50

Here is what your completed table might look like.

trial 1 2 3 4 5 Prediction Total
tally | | | | \cancel{| | | | |} | | | \cancel{| | | | |} \cancel{| | | | |} x x
|
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.

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

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.

III. Write Experimental Probability as Fractions, Decimals or Percents

Just like we could write theoretical probabilities as fractions, decimals or percents, we can also do this with experimental probabilities. If you think about it, it makes perfect sense.

Yes. Sometimes it does. We use percents often in everyday life, so when you are conducting an experiment, it would make sense that you may want to use a percent.

12D. Lesson Exercises

Write each experimental probability as a percent.

  1. \frac{23}{50}
  2. 45 out of 50 flips of a coin, the coin turned up with tails.
  3. .76

Check your work with a friend.

IV. Make Predictions Based on Experimental Probability Given Experimental, Survey or Historical Data

Sometimes you can make predictions based on data that has been collected. When we make a prediction, we have to look at the type of data collected. Sometimes, there will be experimental data because someone will have written a hypothesis and tested the hypothesis. Sometimes, there will be data collected as the result of a survey. Other times, there will be historical data.

You can use data for making predictions about what will happen in the future. The early results from the election for mayor of Bridgewater are shown below. Because this is early data, not all of Bridgewater’s 1500 registered voters have cast their votes.

Precinct 1 2 Total
Diaz 160 116
Green 93 231

Example

If all of Bridgewater’s 1500 voters vote, whom do you predict will win the election and by how many votes?

We can figure out this prediction by following these steps.

Step 1: Total up the votes for each candidate.

\text{Diaz} &= 160 + 116\\&= 276\\\text{Green} &= 93 + 231\\&= 324\\\text{Total} &= 276 + 324\\&= 600 \ \text{votes were cast}

Step 2: Find the fraction of the total votes so far that each candidate got. Change the fraction to a percent.

\text{Diaz} &= 276 \ \text{out of} \ 600\\&= \frac{276}{600}\\&= 0.46 = 46\%\\\text{Green} &= 324 \ \text{out of} \ 600\\&= \frac{324}{600}\\&= 0.54 = 54\%

Step 3: Use the percentage to find out the number of votes each candidate will get out of the 1500 voters. Round the answer to the nearest whole vote, if necessary

\text{Diaz} &= 46\% \ \text{of} \ 1500\\&= 0.46 \cdot 1500\\&= 690\\\text{Green} &= 54\% \ \text{of} \ 1500\\&= 0.54 \cdot 1500\\&= 810

Now you know that according to the early votes, that Green will have more votes if things continue along this trend.

Step 4: Subtract to find the margin of victory.

\text{Green} - \text{Diaz} &= \text{victory margin}\\810 - 690 &= 120 \ \text{votes}

Our prediction based on these early votes is that Green will win the election by 120 votes.

Notice that we can’t be absolutely sure, but that this is a prediction based on the current voting facts!

Real–Life Example Completed

The Juggler

Here is the original problem once again. Reread it and underline any important information.

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, 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.

\text{Probability} &= \frac{successful \ outcomes}{trials \ attempted}\\17:20 &= \frac{17}{20}

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

\frac{17}{20}=\frac{85}{100}=85\%

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

Vocabulary

Here are the vocabulary words that are found in this lesson.

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

Technology Integration

Khan Academy, Probability 1 Module Examples

Time to Practice

Directions: Use what you have learned about experimental probability to complete each problem.

Use the table below 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 x x x x x x

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 below 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 x x x x x

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 below 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 not landing on blue?

A number cube was rolled in a probability experiment 40 times. The results are shown in the table below. Compute the experimental probability of rolling a 2.

dots on number cube \cdot \cdot \cdot \cdot \cdot \cdot

\cdot \cdot

\cdot \cdot

\cdot \cdot \cdot

\cdot \cdot

\cdot \cdot \cdot

\cdot \cdot \cdot

Total
1 2 3 4 5 6
number of times cube landed \cancel{| | | |} \cancel{| | | |} \cancel{| | | |} \cancel{| | | |} \cancel{| | | |} | | | | x
|||| ||| | || |
total from tally 9 8 6 7 6 4
favorable outcomes x x x x x
experimental probability x x x x x x

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 \cdot \cdot \cdot \cdot \cdot \cdot

\cdot \cdot

\cdot \cdot

\cdot \cdot \cdot

\cdot \cdot

\cdot \cdot \cdot

\cdot \cdot \cdot

Total
1 2 3 4 5 6
number of times cube landed \cancel{| | | |} \cancel{| | | |} \cancel{| | | |} \cancel{| | | |} \cancel{| | | |} \cancel{| | | |} x
\cancel{||||} | |||| || \cancel{||||} ||
|
total from tally 10 6 9 7 11 7
favorable outcomes x x 9 7 x x
experimental probability x x x x x x

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 rolling a 2?

Directions: Now look at predictions and hypotheses with experimental probability.

16. Tally results and compute predictions and experimental probability for the coin flip experiment shown in the table below. How close was the actual data to the predicted value?

Hypothesis: A coin lands on heads 50% of the time.

trial 1 2 3 4 5 Prediction Total
tally \cancel{| | | | |} \cancel{| | | | |} | | | | \cancel{| | | | |} \cancel{| | | | |} x x
| ||
heads 6 5 4 7 5 25 27
total flips 10 10 10 10 10 50 50
percent x x x x x 50% 54%

17. Tally results and compute predictions and experimental probability for the coin flip experiment shown in the table below. Double click to check your answers. How close was the actual data to the predicted value? Hypothesis: A coin lands on tails 50% of the time.

trial 1 2 3 4 5 Prediction Total
tally | | | | | | | | | | | \cancel{| | | | |} | | | x x
|||
tails 4 4 3 8 3
total flips 10 10 10 10 10
percent x x x x x

18. The table below shows the results of spinning the spinner 60 times. Compute the predicted values and actual values for the arrow landing on the red section. How close was the actual data to the predicted value?

Hypothesis: The spinner lands on red 25% of the time.

trial 1 2 3 4 5 6 Prediction Actual
tally | | \cancel{| | | |} | | | | | | | | | | --
red 2 5 3 2 1 4 15 17
total spins 10 10 10 10 10 10 60 60
percent 25% 28.3%

19. Use the table to conduct your own coin flip experiment to find the experimental probability of tails coming up in 60 coin flips. Tally and total your results for 60 coin flips in groups of ten.

Hypothesis: A coin lands on tails 50% of the time.

trial 1 2 3 4 5 6 Prediction Total
tally
tails 30
total flips 60 60
percent 50%

20. How many total coin flips did you make?

21. How many flips turned up tails?

22. What percent of your flips turned up tails?

23. How well does your data agree with your prediction?

24. Try another group of 40 flips. Add your results of 40 flips to your previous 60 flips to make 100 total flips. How many flips turned up tails?

^*25. Did your results of 100 flips come closer to your predicted result than your first 60 flips? Explain.

26. Alex Rodriguez compiled this batting record shown over the first 5 months of the 6-month season. Based on A-Rod’s monthly homerun total, predict the number of total homeruns Rodriguez will have at the end of the month of September. [source for A-Rod: ESPN.com]

April May June July Aug Sept
Home runs 14 5 9 7 9

27. If A-Rod’s team plays 1 game each day during the month of September, and his home runs are evenly distributed throughout the month, predict the date on which Rodriguez will hit homerun number 50.

28. At the end of August, A-Rod had 489 at-bats. Predict the total number of at-bats A-Rod will have on the day he hits homerun number 50.

29. Suppose at the end of April you were asked to predict the number of homeruns A-Rod would over an entire 6-month season. What would your prediction be?

30. Suppose Rodriguez finishes the year with a total of 520 career homeruns at the end of 2007. If he continues at his monthly pace from this year, during which year and what month do you predict that A-Rod will hit homerun number 800? (Assume each baseball season has 6 months, April to September.)

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