Mathematics in action: Probability


Dice

Today we begin a series of three posts on probability. We begin with some experiments on rolling dice. To save my wrists, I’ve used random numbers from www.random.org instead.

We begin with individual dice rolls. Rolling 1200 times, we expect each number to come up about 200 times, and the histogram below shows that that’s pretty much what happens. We expect the average (mean) of the numbers rolled to be 3.5, and for my 1200 rolls it was actually very close to that (3.495833). We expect the standard deviation of the numbers rolled to be sqrt(s − 3.52) = sqrt(35/12) = 1.707825, where s is the mean of the squares of the numbers (1, 4, 9, 16, 25, 36). The actual sample standard deviation was 1.704133. It is generally more convenient to work with the variance, however, which is the square of the standard deviation (2.904069, which is very close to 35/12 = 2.916667).


Histogram of 1200 individual dice rolls. The dark blue dots and line show the expected results.

Rolling pairs of dice, we expect a sum of 2 to come up, on average once in 36 rolls. We expect 3 to come up twice (as 1+2 and as 2+1), 4 to come up three times (as 1+3, 2+2, and 3+1), etc. The histogram below shows what actually happens.

We expect the average (mean) of the numbers rolled to be 7, and it was actually very close to that (7.019167). The great thing about adding random numbers is that the variance of the sum is the sum of the individual variances, so we expect the variance to be 70/12 = 5.833333. The actual sample variance for my 1200 rolls was 5.831993.


Histogram of 1200 dice pair rolls. The dark blue dots and line show the expected results.

There is an amusing take on dice pair rolls in Asterix and the Soothsayer, where the soothsayer is trying not to predict the result of a dice pair roll. Given the choice of numbers from I to XII, he guesses VII, which is of course the most likely outcome, and the one which actually comes up (which is the least likely outcome?).

Rolling a larger number of dice, the expected outcome follows a bell curve. The expected mean for twenty dice is 20 × 3.5 = 70 (for my 800 rolls of twenty dice it was actually 70.12). The expected variance is 20 × 35 / 12 = 58.33333 (for my 800 rolls of twenty dice the sample variance was actually 60.42113). The standard deviation (the square root of the variance) determines the “width” of the bell curve. Similar bell curves occur whenever a result is composed of a large number of independent factors (of roughly equal weight) added together.


Histogram of 800 rolls of twenty dice. The dark blue bell curve shows the expected results.


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