Unit 8: Simulating Probabilities
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 Teaching Tips

• It is tempting to use simulations – particularly computer based simulations – to help students "discover" a concept, or to help them understand a concept. This is trickier than it seems. If you are not careful, students will see the simulation as a black box, so the following five-step process might be useful whenever students are working with simulations.
  • First, answer: Why are we conducting the simulation? What question are we investigating? What do we hope to learn?

  • Then determine: On what assumptions is the simulation based? E.g. there are two equally likely outcomes, the population proportion is 60%, etc.

  • Design and conduct the simulation with a “hands-on” experience with the random experiment, for example using a die or a deck of cards. Explain how the assumptions are reflected in the design of the simulation, e.g. heads means a girl, digits 0-5 mean a “yes”, etc. Practice a few simulated trials together, interpret, and plot the results.

  • Predict what will happen.

  • Transition to technology going through the same steps. Explain how the assumptions are reflected in the design of the simulation, run a few trials, interpret and plot.
• Students will not learn by running other people's simulation programs. They must do their own simulations.

• It's very important for students to understand exactly what is happening at each step of a simulation exercise.

• Students must do many different simulations before the idea "sinks" in.

• Students should be able to do a simulation using a random number table AND using a random number generator on a calculator or computer.

• There is no rule for deciding when your approximation of the theoretical probability is "good enough." "Good enough" depends on the context and the accuracy required of the context.

Student Misconceptions and Confusions

• Beware: simulations can be intricate and students tend to lose sight of the goal. It's about estimating a probability (or modeling the distribution of probabilities), not performing an intricate series of taps on a calculator.

• Don't confuse lower case n -- the number of objects in a sample in a single trial -- with N, the number of times you repeat the trial in your simulation. For example, a trial might consist of selecting 100 Californians at random and counting the number who voted for Arnold Schwarzenegger, under the assumption that 60% of the population supported him. To simulate these, we have a bag of six red chips and four black chips. We draw 100 chips with replacement and count the number of reds. We repeat this 5000 times. In this example, n = 100 and N = 5000. (No one said simulations were easy!)

• Some students may wonder what it means for trials to be independent. In the context of a simulation, independent trials means the outcome of any one trial does not affect our assessment of the probability distribution for the outcomes of any other trials.

Resources

• Galton built a famous mechanical simulation: the quincunx. This simulation illustrates the Central Limit Theorem for the binomial distribution and could also be used to calculate approximate probabilities involving adding a series of yes/no random outcomes. You can view a computer version of it here: RAND Central Limit Theorem in Action.  There are many other quincunx displays, but this is the best.

• A classic probability problem that can be fun to simulate with your students is the birthday problem (assuming you have over 30 students in your class.) One possible moral of this problem is that coincidences are more likely than one might think. Which leads us to think that sometimes what we perceive as meaningful patterns are actually simply due to chance. Here is an applet that demonstrates this problem: http://www-stat.stanford.edu/~susan/surprise/Birthday.html