One interesting aspect of programming is having access to libraries and functions that deal with statistics, probability,and randomness.

What I've found for myself when using these libraries is that my innate "feeling" about probability and randomness is way off. In particular, if I generate a succession of random elements, I find that extreme events happen more frequently than my "gut" tells me they should.

I've developed a quick-and-dirty little program to show this. Clicking the Random Number button below will return a random number from a normal distribution having a mean (average) value of 50 and a standard deviation of 20. For those who never took a statistics class (or who have forgotten everything if they did), a normal distribution is quite common in the world: It's the "bell-shaped curve" distribution we're used to seeing. In a normal distribution, the standard deviation value lets us know how "spread out" the distribution is. It also lets us calculate what percentage of values fall within certain ranges.

As noted, the distribution I've created has a mean of 50 and a standard deviation of 20. The chart below shows the relationship between standard deviation and percentiles. 68% of the values in a normal distribution fall within one standard deviation of the mean, and 95% fall within two standard deviations. Thus when you click the button, 68% of the time the random number you get will fall between 30 and 70 (the mean of 50 plus and minus the standard deviation of 20), and 95% will fall between 10 and 90 (the mean of 50 plus or minus 40 [two times the standard deviation of 20]).

I encourage you to take a few minutes and click the Random Number button and watch the results go by. I just did it as a part of checking if this page was functioning correctly. Within 10 clicks I received back a value of -16. This number is more than three standard deviations from the mean. Numbers outside of three standard deviations (that is, numbers less than -10 or more than 110) should come up only 3 times out of 1000. Yet I got one within 10 licks. That's randomness, and it's scary, because as much as we'd like to comfort ourselves with the knowledge that 95% of values are within 2 standard deviations, thus making us "safe" from rare events, that's an aggregate truth, not an individual one. As individuals, we don't get to aggregate; we have to push the button and see what happens, one result at a time.

Give it a try, and see how long it takes for you see a number outside of two deviations (less than 10 or more than 90) or outside of three deviations (less than -10 or more than 110).

Your random number (this time) is: 50