Student Instructions

Unit 5: Lesson 12 - Loops and Simulations

Background

We can often develop intuitions for patterns by performing real-life experiments at a small scale, but verifying our intuitions with larger experiments can prove prohibitively time-consuming, costly, or even dangerous. If we can develop a model or simulation of the phenomenon we wish to investigate, we can confirm our hypotheses much more quickly without the risks or costs associated with a real-life experiment.

Vocabulary

Models and Simulations: a program which replicates or mimics key features of a real world event in order to investigate its behavior without the cost, time, or danger of running an experiment in real life.

Lesson

• Calculate statistics from a coin flipping experiment.
• Create a simulation to repeat this experiment for much larger inputs.
• Create updated hypotheses based on the results of simulations.

Resources

• Activity Guide: Flipping Coins (PDF | DOCX)

Continue

Student Instructions

Make a Hypothesis

Over the next several exercises we are going to be building a program that simulates flipping coins in order to find out how many flips it takes to get 10,000 heads in total and how many flips it takes to get a streak of 12 heads in a row. If you're following along on the worksheet, you might have already tried these experiments at a smaller scale, flipping a coin until you got 5 total heads and a streak of 3 heads in a row. Doing this smaller scale activity may have given you some intuition for what the outcome of the larger experiment will be, but before running an experiment it's usually a good idea to develop a hypothesis.

What will be the maximum and minimum number of flips it will take to get 10,000 heads? When trying to get 5 heads some groups may have taken exactly 5 flips while others took 15 or more (3 times the number of heads you were looking for!). Do you think this wide range of values will persist when looking for 10,000 heads?

What will be the maximum and minimum number of flips it will take to get a streak of 12 heads? Again think about the patterns you've observed. If you need 4 times as many heads in a row will it take 4 times as long? Try to justify the values you choose.

Write down your predictions on the worksheet so you can reference them later.

It's not important whether you're right (you'll have the answer soon enough anyway!) but it will let you reflect on your thinking. In fact, you might even get some new ideas you'd like to investigate as a result!

Continue

Student Instructions

Starting Small

To start, we're going to simulate flipping a coin 10 times. You might be thinking that isn't many coin flips, and that we could just do those flips in real life, but this is actually an important step in developing a simulation. At small scales we can make sure our code is working as intended because we can still visually confirm its output. Once we're convinced that the logic of our program is reliable we'll move up to simulating larger numbers of flips.

The core logic of our program will be focused on a while loop that simulates flipping a coin by repeatedly generating random 0's or 1's using randomNumber. This is a great opportunity to keep practicing using loops while applying your knowledge of variables, iteration, and if statements.

Do This:

• When we want to flip a coin with a computer we will instead generate a random number between 0 and 1.
• Write a program that uses a while loop to flip a coin 10 times and writes the value of each flip to the screen. The example below shows how your program should run.
• HINT: you will need to use a counter variable in your while loop to keep track of how many times the coin has been flipped.

Student Instructions

Counting Heads

Let's say that a 1 is a heads. If we want our simulation to run until we reach a certain number of heads then we will need a way to count the number of heads that have been flipped. In order to do this you will need to add a variable that acts as a counter. Initialize it to 0, and every time you flip a heads (1) increment your counter by 1. At the end of your program you should write the value to the screen.

Do This:

• Add a variable that counts the number of heads (1's) and write its value to the screen after each flip.
• Run your program severals times and validate that the number of heads recorded is correct by visually checking the flips you've printed to the screen.
• Hint: you will also need to store your current coin flip in a variable to complete this challenge.

Student Instructions

Changing the Loop Condition

Currently our loop condition is based on a counter variable that keeps track of the total number of flips, but our simulation should only run while we have fewer than the target number of heads. In this exercise you are going to change the condition used by your while loop so that your simulation terminates once you have flipped 5 total heads. At the end of the loop we will write the total number of flips to know how many flips it took to get 5 heads.

Do This:

• Change the looping condition to use the variable you are using to keep track of the number of heads. Your loop should run as long as you have fewer than 5 total heads.
• Add code after your loop that writes the total number of flips.
• Test your program several times to make sure it works as expected. See the example below.
• Note: the simulation is always terminating once it flips its 5th heads. Sometimes this means the text output runs off the screen. We will address this in the next exercise.

Student Instructions

10,000 Heads!

You're ready to increase the number of heads your simulation is looking for and test your first hypothesis. Before we move up to the full 10,000 heads, however, we're going to perform a quick check of our program logic. When you make changes to your program it is possible that some portion of your programming logic will stop working as you expected. In order to feel more confident about your model you will first change the number of heads you are looking for to a number that we can still visually verify (7 heads). If our code still works after making changes then we should be confident that it should work at 10,000. We will remove the intermediate output and run the full simulation!

Do This:

• Increase the number of heads you are looking for to 7 and visually confirm that the code is still working as you expect.
• If everything seems to be working, comment out the write command that displays the results of each flip. You can use to comment out a single line of code. Make sure that the total count of flips still prints though!
• Run your simulation and find out how long it takes to get to 10,000 heads. Then run it a few more times. What patterns are you noticing? Record your results and move on.

Student Instructions

Updating Your Hypotheses

If everything has gone well you should already be able to make a determination about your first hypothesis.

• How close were your guesses on the high and low bounds for the number of flips to get 10,000 heads?
• Were you at all surprised by this result? If they were different from what you predicted why might this be the case?

Take some time to write out your thoughts and whether your understanding of this system has changed. Update either of your hypotheses to reflect what you observed in this first experiment.

Continue

Student Instructions

Streaks of Heads

We are going to alter our simulation so that it doesn't count the total number of heads, but rather the longest streaks of heads. This will allow us to simulate how many flips it takes to get 12 heads in a row.

To begin with you will change your looping condition so that the loop again only runs 10 times. This will allow us to visually confirm our code is working.

Keeping Track of Streaks: We know we need to count streaks of heads, how do we do this in code? Do we need to keep track of all the previous flips so we know that we're on a streak?

The answer is: no. We can instead just count in a clever way that makes our code pretty simple. Make a variable to use as a counter and...

• every time you see heads, add 1 to a counter.
• every time you see tails, set the counter back to 0.

Here is some psuedocode showing how it works. You might take a minute to study and reason about why and how it works.

Pseudocode

// Ouside loop
headsCount <-- 0

// Inside loop  
IF (current flip is a heads)
  headsCount <-- headsCount + 1
ELSE
  headsCount <-- 0
DISPLAY (current flip)
DISPLAY (headsCount)

Do This:

• Change your while loop's condition so that it only runs 10 times.
• Add code to the simulation that displays the length of each streak of heads following the logic described above. Note that pseudocode should translate to JavaScript pretty easily.
• The example below shows an example of what output from your program might look like.

Student Instructions

Changing the Loop Condition: Streaks of Heads

We want our simulation to run while the streak of heads is less than a target length. In order to do this we'll need to change our looping condition to use the variables we've been using to count our streak of heads. To begin with your simulation should look for a streak of 3 heads so that you can still visually confirm the output.

Do This:

• Change the condition used by your while loop so that it now runs while the streak of heads is less than 3.
• Test your program by running it and visually confirming that its output is correct. It should run like in the example below.
• Note: As before it's possible that your output will sometimes run off the screen. You can still confirm your code is working as you intend (it should never run after a streak of 3) and we will address this issue in the next exercise.

Student Instructions

Streaks of Heads: 12 in a Row

We're almost ready to test our second hypothesis and find out how long it takes to get 12 heads in a row. Just as before we're first going to visually test our code with a different length streak (4) before removing most of the visual output and running the code for a streak of 12.

Do This:

• Change your simulation to run until you get a streak of 4 and visually verify that the values calculated are accurate.
• Comment out the lines of code that write the current coin flip and the current streak count. You can use to comment out a single line of code.
• Change your simulation to run until you get a streak of 12 heads.
• Run your simulation a few times and record your results!

Student Instructions

Updating Your Hypotheses - Part 2

You should now be ready to compare the results of your simulation to your second hypothesis.

• How accurate was your prediction of the upper and lower bounds for the number of flips to get a streak of 12 heads?
• Were you at all surprised by this result? What might explain the differences between your prediction and what you observed?

Just as before, you should update your hypotheses to reflect the results of your simulation. Try to come up with a reasonable explanation for the results you observed.

Continue

Student Instructions

Keep Experimenting!

Your simulation can be updated or changed however you like. What new questions do you have? Do you want to look for longer streaks of heads? Do you want to roll a die instead of flipping a coin? Make any changes and improvements to your simulation that you like.