11.8: Probability and Simulations
Introduction
The Bike Selection
Telly stood in front of the window and looked at the bicycle. It was perfect. She had debated all of the different options that she could have and still she thought that the one in the window of the bike shop was the perfect one for her.
“I have to have it,” she said smiling.
“I guess you made up your mind,” Carey said.
“Yes, imagine if we had asked all of our friends which bike I should get, we would have had a ton of different answers,” Telly said.
Telly is correct there would have been a lot of answers to keep track and tally.
This is your task. Design a simulation where Telly would have surveyed her classmates and figured out the probability that she would have selected this bike out of the other 16 options.
Pay attention through this lesson and you can figure this out at the end of the lesson.
What You Will Learn
In this lesson, you will learn how to complete the following skills.
 Perform directed simulations to explore experimental probability using concrete objects.
 Perform directed simulations to explore experimental probability using technology.
 Select and use different models to simulate events.
 Make and compare predictions based on theoretical probabilities and related simulations.
Teaching Time
I. Perform Directed Simulations to Explore Experimental Probability Using Concrete Objects
You can test theoretical probability calculations by performing a simulation. A simulation is a way of collecting probability data using actual objects, such as coins, spinners, and cards. Let’s look at an example.
Example
Conduct a simulation to see how many times heads comes up when you flip a coin 50 times.
Step 1: Make a table like the one below. Conduct your simulation in groups of 10 flips. Leave lots of room to tally your results. Write in a prediction of how many times heads will turn up.
trial  1  2  3  4  5  Prediction  Total 

tally  x  x  
heads  25  
total flips  50 
Step 2: Begin conducting your simulation. Tally your results. Some sample data is shown.
Remember, this is a sample and not actual data.
trial  1  2  3  4  5  Prediction  Total 

tally  \begin{align*}{}\end{align*}  \begin{align*} \cancel{}\end{align*}  \begin{align*}{}\end{align*}  x  x  
heads  25  
total flips  50 
Step 3: 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*}  x  x 
heads  4  5  3  6  5  25  23 
total flips  10  10  10  10  10  50  50 
Now that you are set up and understand the process, run the simulations and solve the problems that follow. Record your data. Make sure you make a table and predictions for each simulation.
Simulation 1:
Run a simulation of 60 coin flips to see how frequently tails turns up.
 How many times did you predict tails would occur? How many times did it actually occur?
 How well did your data agree with your prediction? Explain.
Work on this simulation with a partner. Record your data and then discuss your answers.
II. Perform Directed Simulations to Explore Experimental Probability using Technology
Many calculators and websites have random number generators and other features that can be used to generate data for simulations. Here we’ll use the website http://www.random.org to run a coin flipping simulation.
Step 1: Make a table for collecting coin flip data like the one shown. Fill in your prediction. You won’t need to tally data here – the computer will do it for you.
trial  1  2  3  4  5  6  7  8  9  10  Prediction  Total 

heads  
total flips 
Step 2: Open the webpage http://www.random.org/. Click on “coin flipper.” Here we’ve selected 10 as the number of coins to flip on each trial. Click on “Flip coin(s)”. Record the data in the table. Sample data collected from the website is shown below.
trial  1  2  3  4  5  6  7  8  9  10  Prediction  Total 

heads  6  4  6  2  7  6  6  5  5  5  50  52 
total flips  10  10  10  10  10  10  10  10  10  10  100  100 
Step 3: Analyze your data. How accurate was your prediction?
Use the website http://www.random.org/ (or some other simulator) to run the simulations and solve the problems that follow. Record your data. Make sure you make a table and predictions for each simulation.
Simulation 1:
Run a simulation of 100 coin flips to see how frequently tails turns up. Use a recording table like the one shown above.
 How many times did you predict tails would occur? How many times did it actually occur?
 How well did your data agree with your prediction? Explain.
 Try another 100 flips. How did the additional data change your results? Explain.
Your answers will vary. Work with a partner and figure out the results for each simulation. Be sure to record your data.
III. Select and Use Different Models to Simulate Events
When working on different types of problems, we have to work to figure out what model makes the most sense. You now know how to use a website, create a table and perform experiments. Deciding which one to use will always be dependent on your own choice. Sometimes, you will choose one method and then figure out that you really need a different one. Let’s look at how we can make predictions based on theoretical probability.
IV. Make and Compare Probabilities based on Theoretical Probabilities and Related Simulations
In the real world, probability experiments don’t always turn out to have ideal results. When you flip a coin 10 times, the “ideal” result would be 5 heads and 5 tails. But you know from experience that with such a small number of trials anything can happen. Change the number of trials to 20 or 1000 and you might see results closer to ideal. Do 500 or 1000 flips and you would expect your results to be closer.
Theoretically, it seems clear that additional trials give results that are closer to ideal. But keep in mind that in the real world ANYTHING can happen. To test to see if large numbers of trials really give results that are closer to ideal, let’s run some random number simulations using large numbers of trials.
Step 1: Predict how many 1s, 2s, 3s, 4s, and so on would appear if 1000 random numbers from 0 to 9 are generated. Record your predictions in a table like the one shown. Fill in the table.
digit  0  1  2  3  4  5  6  7  8  9  Total 

prediction  100  100  100  100  100  100  100  100  100  100  1000 
group 1  1000  
group 2  
group 3  
group 4  
total 
Step 2: Open the webpage http://www.random.org/. Click on “random integer generator”.
Step 3: To get 1000 random numbers from 0 to 9:
 Fill in: “Generate 1000 random integers.”
 Fill in: “Each integer should have a value between 0 and 9.”
 Fill in: “Format in 10 column(s).” (You can format differently if you like.)
Step 4: Click on “Get numbers” and obtain your 1000 random integers.
IMPORTANT: If you don’t want to spend all day counting, do the following:
Step 5: Highlight and COPY all 1000 numbers on the webpage.
Step 6: PASTE the numbers into a wordprocessing document. You may need to do a little bit of reformatting, but if things go well you’ll end up with a list of numbers like the one below.
Step 7: Now highlight all of the numbers on the list (in your document, not on the webpage) and go to the FIND function of your word processor. You can use the FIND function to count your data so you don’t have to do the work! (Note: different computer systems work differently, so you may need to modify these directions for your own system.)
 To find the number of 0s on the list, go to the FIND function, type in “0” and click “highlight all items for this selection only” or its equivalent. The results for our sample data list showed the number of 0s was 94. You, of course, will get different results.
 Record this number on your table.
 Repeat the process to find the number of 1s, 2s, 3s, 4s, 5s, 6s, 7s, 8s, and 9s in a similar way. Record your data as group 1 in your table. The table below shows typical data from the list we generated below.
digit  0  1  2  3  4  5  6  7  8  9  Total 

prediction  100  100  100  100  100  100  100  100  100  100  1000 
group 1  93  88  100  110  102  94  98  104  100  111  1000 
group 2  
group 3  
group 4  
total 
Step 8: Repeat the process to create group 2, group 3, and so on. In a short time, you can collect thousands of data trials.
You can think about what this model allows us to do. We can use technology and objects too to calculate results.
Step 9: Total up your data and compare it to ideal results. For example, in the data we collected, the digits 2 and 8 turned out to agree with the theoretical value of 100.
Now let’s go back to the problem from the introduction.
RealLife Example Completed
The Bike Selection
Telly stood in front of the window and looked at the bicycle. It was perfect. She had debated all of the different options that she could have and still she thought that the one in the window of the bike shop was the perfect one for her.
“I have to have it,” she said smiling.
“I guess you made up your mind,” Carey said.
“Yes, imagine if we had asked all of our friends which bike I should get, we would have had a ton of different answers,” Telly said.
Telly is correct there would have been a lot of answers to keep track and tally.
This is your task. Design a simulation where Telly would have surveyed her classmates and figured out the probability that she would have selected this bike out of the other 16 options.
Now design your survey with a friend.
Solution to Real – Life Example
Answers will vary. Use this as an opportunity to present different simulations and discuss them with your friends.
Vocabulary
Here is the key vocabulary word that is found in this lesson.
 Simulation
 a way of collecting data using objects such as spinners, coins or cards.
Time to Practice
Directions: Run a simulation using playing cards. Make a stack of the Ace, Jack, King, Queen, and Ten of each suit. Predict how frequently an Ace will turn up. Run 60 trials in groups of 10. Return the card to the deck and shuffle after you choose each card. Use a table like the one below.
trial  1  2  3  4  5  6  Prediction  Total 

tally  
Ace  
total tosses  10  10  10  10  10  10 
 How many times did you predict an Ace would occur? How many times did it actually occur?
 How well does your data agree with your prediction?
Directions: Run a simulation of 72 number cube tosses in groups of 12 to see how frequently 4 or 5 turns up. Use a table like the one below.
trial  1  2  3  4  5  6  Prediction  Total 

tally  
4 or 5  
total tosses  12  12  12  12  12  12 
 How many times did you predict 4 or 5 would occur? How many times did it actually occur?
 How well does your data agree with your prediction?
 Try another group of 36 tosses. Add your results of 36 tosses to your previous 72 tosses to make 108 total tosses. How well did your data now agree with your prediction? Explain.
Directions: Use http://www.random.org for each simulation. Run a number cube simulation to see how many times each number on the cube comes up. In http://www.random.org/, click on the link that reads “dice roller”. Choose 12 for the number of number cubes (dice) you want to roll. Set up a table like the one below to have a total of 96 rolls.
Tally up the number of 1s, 2s, 3s, 4s, 5s, and 6s that turn up and record them in the table. Keep rolling until you have a total of 96 rolls.
number 
1

2

3

4

5

6

Total 

tally  
total  96  
prediction  16  16  16  16  16  16  96 
 How many times did you predict each number on the number cube would appear in 96 rolls?
 How many times did it actually appear?
 How well does your data agree with your prediction?
 Try an additional 96 rolls. How did the additional data change your results? Explain.
Directions: Run a random number simulation to see how many times each number from 0 to 9 appears in 100 numbers. On http://www.random.org/, do the following:
 Click on “random integer generator.”
 Choose 100 for “Generate 100 random integers.”
 Choose 0 and 9 for “Each integer should have a value between 0 and 9.”
 Click on “Get numbers.”
Record your data in a table like the one below. Make a prediction for how many times each digit will turn up out of 100 total digits.
digit  0  1  2  3  4  5  6  7  8  9  Total 

tally  
total  100  
prediction  10  10  10  10  10  10  10  10  10  10  100 
 How many times did you predict each digit appears?
 How many times did it actually appear?
 How well does your data agree with your prediction?
 Try an extra 100 digits. How does the additional data change your results?
Directions: Go back to the data you collected in the last section on theoretical probability and predictions. Use that data to answer the following questions.
 How does your data compare to ideal results?
 Which digits were closest to ideal?
 Which were farthest?
 Try pooling your data with other students. You may be able to get 10,000 or even 100,000 data trials. As you collect more data, how close do your values get to ideal?
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