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# Simulation of Random Behavior and Probability Distributions

## Conduct experiments to evaluate predicted probability

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Practice Simulation of Random Behavior and Probability Distributions
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Use Simulations to Explore Experimental Probability

Have you ever tried to select a new bike? Take a look at this dilemma.

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. Use this Concept to help you.

### Guidance

A simulation is a way of collecting probability data using actual objects, such as coins, spinners, and cards.

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?

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.

Answer each question true or false.

#### Example A

A simulation is an experiment.

Solution: True

#### Example B

To explore experimental probability, you could simply spin a spinner one time.

Solution: False

#### Example C

To use a coin in a simulation, you would need to flip the coin many, many times.

Solution: True

Now let's go back to the dilemma from the beginning of the Concept.

To accomplish this task, Kelly could write out all of the 16 options for bicycles. Then she could go around and ask her friends to rate the bikes 1, 2 or 3. One being the first choice etc.

Then Kelly could calculate her scores and narrow it down to the bikes that received the most votes.

The bicycle that was selected the most times would be the winning bike.

### Vocabulary

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

### Guided Practice

Here is one for you to try on your own.

Jessie decided to conduct an experiment with a spinner. The spinner is divided into four colors: red, blue, orange and green. Jessie predicted that out of 30 spins that the spinner would be red 10% of the time.

She conducted the experiment and the spinner was red four times.

Is her prediction correct?

Solution

To figure this out, we must first write a probability and then compare it to the 10% that Jessie predicted.

The actual result is that the spinner was red 4 out of 30 times.

We can write that as a fraction.

430\begin{align*}\frac{4}{30}\end{align*}

Now let's convert it to a percent.

.133=13.3%\begin{align*}.133 = 13.3\%\end{align*}

Jessie's prediction was too low. The actual result was higher than 10%.

### 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 using playing cards. Make a stack of the Ace, Jack, King, Queen, and Ten of each suit. Predict how frequently a King, a Queen and a Jack 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 King would occur? How many times did it actually occur?
2. How many times did you predict a Jack would occur? How many times did it actually occur?
3. How many times did you predict a Queen 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: Now design your own simulation and use a spinner or two number cubes.

### Vocabulary Language: English

Simulation

Simulation

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