Topics Probability

Question Am I correct in thinking that the definition of randomness is that all possible outcome had an equal chance of occurring? And that in an event being totaly random, absolutely anything could happen? The likeliness of a banana peeling itself open is the same as a whole new universe, the size of a basketball, appearing is the exact same? Thank you for your time. ~Kris S.

Accepted:
January 28, 2006

Comments

That is not one of the definitions of "randomness" with which I am familiar.

Rather than consider how to define "randomness", let's consider the idea of every possible outcome having an equal chance of occurring. Suppose I ask you to select a prime number "at random". What would it be for every possible outcome to have an equal chance of being selected?

One way to understand this would be for every prime number to have the same chance of being selected. But there are an infinite number of prime numbers. So the only way for each prime number to have the same chance of being selected is for each to have zero chance of being selected.

However, it would then seem reasonable to conclude that the chance of your selecting *any* prime number at all is the chance of your selecting the first prime number (which is 1) plus the chance of your selecting the second prime number (2) plus the chance of your selecting the third prime number (3) plus the chance of your selecting the fourth prime number (5) plus the chance of your selecting the fifth prime number (7) and so forth for all of the prime numbers. That's a sum of zeros (albeit an infinite number of zeros), which is zero! But you were instructed to select a prime number! I would have thought that your chance of doing so was 1 (i.e., 100%).

Let's try that again. We were looking at the idea that every possible outcome has an equal chance of occurring. Now one possible outcome is your selecting a prime number greater than 100. Another is your selecting a prime number less than 100. If every possible outcome has an equal chance of occurring, then these two outcomes have the same chance: 50% each. Is that right? There are infinitely more prime numbers above 100 than below. Perhaps, then, we should say that there is an infinitely greater chance of your selecting a prime number above 100 than below. So your chance of selecting a prime number above 100 is 1 (that is, 100%) and your chance of selecting a prime number below 100 is 0.

By the same reasoning, of course, there is zero chance of your selecting a prime number below 200. And there is zero chance of your selecting a prime number below 300. And for any finite number n, there is zero chance of your selecting a prime number below n.

For that matter, if we say that there is a 50% chance of your selecting a prime number below 100 (and a 50% chance of your selecting a prime number above 100), then by the same reasoning, we should say that there is a 50 % chance of your selecting a prime number below 200 (and a 50% chance of your selecting a prime number above 200). But then it would seem that there is 0% chance of your selecting a prime number between 100 and 200, which seems unfair to them.

The lesson here is that dividing chances equally among possible outcomes is a subtle business. For one thing, it is subtle when there are infinitely many outcomes. That infinity raises problems. For another thing, it is subtle when there are different ways of characterizing the possible outcomes (such as 1-100 and greater than 100, on the one hand, and 1-200 and greater than 200, on the other hand).

Here's another famous example of the latter point. Suppose you take a trip and your average speed will be between 30 and 60 miles per hour. What is the chance that your average speed will be less than 45 miles per hour? You might be tempted to say: 50%. But 30 miles per hour is 2 minutes per mile, and 60 miles per hour is 1 minute per mile. So presumably, you should say that there is a 50% chance of your taking on average less than 1.5 minutes per mile, and a 50% chance of your taking on average more than 1.5 minutes per mile. The trouble is: 45 miles per hour is not 1.5 minutes per mile!