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# 11.8: Probability and Simulations

Difficulty Level: At Grade Created by: CK-12

## 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.

“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
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
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.

1. How many times did you predict tails would occur? How many times did it actually occur?

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
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

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.

1. How many times did you predict tails would occur? How many times did it actually occur?
3. 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 web-page.

Step 6: PASTE the numbers into a word-processing document. You may need to do a little bit of re-formatting, 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 web-page) 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.

## Real-Life 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.

“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
1. How many times did you predict an Ace would occur? How many times did it actually occur?

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
1. How many times did you predict 4 or 5 would occur? How many times did it actually occur?
3. 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
1. How many times did you predict each number on the number cube would appear in 96 rolls?
2. How many times did it actually appear?

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
1. How many times did you predict each digit appears?
2. How many times did it actually appear?
4. 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.

1. How does your data compare to ideal results?
2. Which digits were closest to ideal?
3. Which were farthest?
4. 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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Date Created:
Jul 25, 2013
Apr 11, 2016
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