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Grade 7:  The Learning Equation Math

42.03 Monte Carlo Simulations

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Chance & Uncertainty

Refresher pp 92-93

Learning Outcomes:

The student will:

Monte Carlo Simulation

Tossing a Coin


Monte Carlo simulation is particularly useful when you want to predict the overall outcome of a series of related events when you only know the statistical probability of the outcome of each component event.

The simplest such situation must be the tossing of a coin. Any individual event will result in the coin falling with one side or the other uppermost (heads or tails). However, common sense tells us that, if we tossed it a very large number of times, the total number of heads and tails should become increasingly similar. We could simulate this by using a random number between 1 and 100 to decide which side would be uppermost, giving a 50% chance of it falling on either side by making heads any number below or equal to 50 and tails any above 50.

This element of chance at each event is where the name Monte Carlo, synomymous with games of chance, comes from.

The working simulation

Here is a working example of the coin-tossing simulation with lots of information in addition to which side the coin lands.

Click on one of the option buttons to select how many times you want the coin tossed - I have limited the range to 100 000 times simply because Javascript is not too fast.

If you select the whole series option, the program will work through 100,000 tosses and will eventually display the running totals it recorded for the smaller samples on the way.

TURNS HEADS TAILS RATIO
tosses % % tosses heads/tails

Source and permission for noncommercial use:  Monte Carlo Introduction

 

Monte Carlo

 

Monty Hall Simulation

Please be patient while the program loads. If part of it disappears while running, jiggle the window size. Close this window when you are done.

 

Sorry, but you need a Java-enhanced browser to use this simulation.

freeware source:  Monty Hall Simulation

Monty Hall Simulation

Pretend you are the contestant. You are told there is a grand prize behind one of three doors and that you will win that prize only if you select the correct door. When you select a door, the contest host opens a door that does not contain the grand prize. The host offers you the option to switch to the remaining door or stick with the door you chose at the beginning of the game. What should you do?

Sorry, but you need a Java-enhanced browser to use this simulation.

freeware source:  Monty Hall Simulation

Think about what you know.

  • Initially the probability of chosing the door wth the grand prize behind it is 1/3, which means there is 2/3 chance the prize is behind one of the remaining doors.
  • When the host opens one of the remaining doors, we know the grand prize is not behind that door. The probability of the grand prize being behind the door you selected is still 1/3. The probability that is behind the remaining door, must be 2/3.

You will win most often if you switch! Test it out by running the simulation above. Choose a door by clicking on it. One of the remaining doors will open, revealing a pig. You may stay wih your original selection or switch to the the remaining door. Calculation of the experimental probability of winning with the original door and the experimental probability of winning by switching doors are prvided.

 

If you want to see the results of a number of trialss without having to complete each game try:

Simulation: http://math.ucsd.edu/~anistat/chi-an/MonteHallParadox.html

  • Number of Games:  choose at least 20.
  • Change the arrow: click on "ALWAYS SWITCH" or "NEVER SWITCH".

Do your experimental results agree with the probabilities given in the cell before this simulation?  Does the experimental probability show more success when the "ALWAYS SWITCH" was selected?

Dice

Lottery

 

Review:

Monty Hall, Three Doors - individual trial applet

Monty Hall - multiple trials applet

The Let's Make a Deal Applet

 

Enrichment:

Buffon's Needle

Cereal Boxer

Picking Fortune Cookies

 

Key Terms: A-E F-J K-O P-R S-Z

outcome, simulation

Prerequisite Skills:

Exploring Probability:  Grade 7 Lesson 42.01

Exploring Probability:  Grade 7 Lesson 42.01

Exploring Probability:  Grade 7 Lesson 42.01

Exploring Probability:  Grade 7 Lesson 42.01

Reading and Interpreting Graphs:  Grade 7 Lesson 41.03

 

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Comments to:  Jim Reed
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