A Dice Ensemble¶

This worksheet is designed to study the difference between an experimental ensemble (related to experiments) and a statistical mechanical ensemble for rolling two $N$-sided dice (and only considering the sum).

This will allow you to choose $n$ for an $n$-sided die (the default is six), and the number of times you “roll” it ($N$, which defaults to 1000). Skip down to the “Run calculations” section if you just wish to test the code. In that case, just change what you see in the input that has “run_experiments(n=6,N=1000)” in it, by changing the numbers after the = for $n$ and $N$. Then click “Run All” in the Cell menu above (if you are using CoLab, this is in the Runtime menu). After you have run the notebook once, you can then add lines below and hit “shift-enter” to run other experiments.

Things to try out:

• Run the “experiments” for a fixed $n$ and increasing $N$. Tabulate the results and see how they approach the theoretical probability (which is shown in the last column).

• Go to the “Functions” section and determine how to make each outcome occur with a different probability. Currently all outcomes are equally likely.

Functions¶

In this section, we have imported libraries that we need in order to simulate these outcomes. If you’re only interested in playing around with the results, just “run” the entire notebook and move to the next section. If you’re interested in coding and wish to modify this, have at it!

Run calculations¶

Type

run_experiments( )

and hit Enter to test this out. If you do this, you’ll be asked for the number of sides of the dice ($n$) and the number of tota rolls ($N$). Alternatively you could type

run_experiments(n = 6, N = 100)

to simulate flipping 100 ordinary dice.

You will then see the results showing how many times each allowed outcome occurs with the fraction of total outcomes, as compared with the theoretical probability that should occur. Note in all cases, we consider all individual outcomes for each die equally likely.

Test out the same setup (same number of outcomes) for larger and larger values of $N$, to see that the experimental and theoretical results agree.