(Archive Question of the Week)
Having recently discussed a couple different issues that touched on the relationship of math to reality, I was reminded of this old favorite – a question that is not asked often enough, and reveals some of the “dirty little secrets” behind math. Math is not reality; it is often used to model reality, but as such is usually just an approximation. We need to determine in some way whether the math we use is applicable. Or, to put it another way, math has its own reality, which may or may not match up with the uses to which we put it.
Recall that probability is defined, at a basic level, by counting the number of equally likely outcomes (such as the six numbers you can get by rolling a die), and dividing the number of “successful” outcomes by the total number of outcomes. It is important that you know the outcomes you count are in fact equally likely.
But how do you know whether they are?
Here is the question, from Chris in 2008:
How Do You Know That Events Are Equally Likely? How do you determine whether the events of a problem are equally likely? I can't seem to find any information regarding how to determine if events are equally likely. Most texts don't get into it at all and others use circular logic to show it. For example, the events of a fair coin toss are equally likely because they each have a probability of 1/2. But you can only use that calculation once you have determined that the events are equally likely. How do you make that determination?
Textbooks probably gloss over this because they think it would be too complicated to explain, and perhaps because it might shake students’ confidence in math. But I think it is important to have an accurate picture of where math fits in. I gave three different answers to the question.
We don’t KNOW at all; we ASSUME
You may have noticed that in many math problems they will say "Assume the coin is fair", or something like that. This may seem like cheating, but it is really what math is all about: reasoning from "axioms" (basic assumptions) that define the subject of our reasoning. Whether the assumption makes sense is the subject of the second answer, but we're not ready for that yet! In math, we commonly start with a model of some concept in the world, such as a fair coin or a flat surface; rather than deal with all the complexities of real coins or surfaces, we think about what would be true of an IDEAL coin (heads and tails are equally likely, and there's no other option like standing on edge) or plane (you can draw one line through any two points, etc.). Then we reason based on those assumptions, so that all our conclusions will be definitely true IF those assumptions are true.
Math deals with idealized situations. We want that coin to be “fair”, because that makes it easier to work with; so we assume it is when we do our calculations – creating for ourselves this ideal imaginary world in order to reason about it. Math can’t work without assumptions, because it is essentially nothing more than logical reasoning applied to assumptions (axioms).
We don’t know anything PERFECTLY; we APPROXIMATE
Any real coin is likely to be a little biased in one direction or the other; and there will not be exactly as many boys born as girls. But experience tells us that it is reasonable to assume, for many purposes, that an ordinary coin is likely to be very close to fair, and that one is about as likely to have a boy as a girl. So we make those "simplifying assumptions" when we don't need extreme precision in our answers. Sometimes we do want to be really sure of our answers (in the real world--maybe because real money depends on it); then we don't just go by general EXPERIENCE, but by careful EXPERIMENT. We toss a thousand coins thousands of times each under controlled conditions and determine just how close to fair the average coin is, and how far from fair any given coin is likely to be. This field of study is called statistics, and it can provide the basis for exact calculations of real probabilities--as far as we know, and as long as the population of coins or children we are working with matches the one we studied. Again, we can never be exactly sure ...
We want our results to be reasonably close to reality, so we choose to make reasonable assumptions. So the ideal world we work in is intended to be a good model of the real one. When it matters, we check whether the results of our assumptions match with observation — everyday experience when that’s enough (as in a classroom), and precise experiment when we need more assurance (as in a lab).
Use stated assumptions or common sense
The third answer is that in problems you are given, you are expected to choose equally likely outcomes based on a combination of standard assumptions that have been presented in your text or elsewhere, common sense, and your knowledge of probability. For example, when you toss two coins, you either have been told to assume they are fair coins, or you know from experience or from statistics that they are close to fair, so it makes sense to consider heads and tails on EACH INDIVIDUAL coin as equally likely. You also know enough about probability to realize that that assumption would conflict with the easier assumption that "no heads", "one head", and "two heads" are equally likely. In particular, you have learned that compound events such as this can't be assumed to be equally likely, but simple events (like a single coin) often can. This becomes a sort of intuition: you've seen it happen enough that you will be much more willing to assume that simple events are equally likely than that more complicated things are. You break things down to the simplest possible parts, and then decide whether it makes sense to suppose that those are equally likely.
So in a problem, if you are not directly told what outcomes are equally likely, you go to the simplest events you can find (those from which compound events can be built), and ask yourself whether common sense, experience, or previous statements in the book make it reasonable to assume it. (The assumption that no heads, one head, and two heads are equally likely is wrong, because if each coin is fair, then outcomes HH, HT, TH, and TT are equally likely, so that the probability of one head is 2/4 = 1/2, not 1/3.) In real life, where it is not a teacher but your results that will judge your correctness, you would look at statistics to confirm your assumptions.
One more comment:
I should note one other thing: It is possible to solve problems WITHOUT any equally likely outcomes, and higher level study of probability does just that. The idea of equally likely outcomes just makes it easy to explain the basic concepts, and to solve problems. If you are told that a coin had a probability of 0.30 of heads and 0.70 of tails (as for that spinning penny), you can just take THAT as your assumption, with no actual equiprobable events in sight.
So, the initial question came from a context where probability is defined in terms of equally likely events; but once we get past the basics, we can work with probabilities in a more general way, starting with axioms that do not require equally likely outcomes. This gives us a lot more flexibility. For an introduction to that, see this page:
Probability Axioms and Theorems
To close, I want to look at an answer to another question (from Kelly in 2010) that relates to the same topic:
Equally Likely vs. Equally Possible When a baby is born, it is either right-handed or left-handed. Are these possibilities equally likely? From what I know, most babies are right-handed, but it seems there is still a 50/50 chance they could be born either. I'm actually a parent trying to understand my son's homework. There are several examples that seem clear enough, like rolling a number cube. But another one says that the Pittsburgh Steelers play a game -- and either win, lose, or tie. As with the question about which hand will be a baby's dominant one, it seems "unlikely" that a professional football game will end in a tie; but the question is to decide and explain whether the possible results are "equally likely." I can't explain this.
This is an example of how textbooks try to help students understand the importance of equally likely outcomes in the (elementary) definition of probability. It is meant to relate to common sense, but I imagine a lot of people, like Kelly, feel uncomfortable using common sense in talking about math. What do we really mean by equally likely? Sound familiar?
I gave a brief answer, and referred to the discussion above:
"Equally likely" is sometimes implicit in our definition of a problem (like the die, or "number cube," which is designed to make each outcome happen as often as any other), and sometimes dependent entirely on our empirical observations. It's conceivable, I suppose, that a pro football team might win 1/3 of their games, lose 1/3, and tie 1/3; but certainly there is no reason to expect that! "Equally likely" is about what we expect, not what might happen. The main idea here is to make students stop and think before they use a set of outcomes as the basis for probabilities. The mere fact that there are 3 possible outcomes does not mean that each has probability 1/3; you need to have some basis for supposing that this is true. So don't overthink the questions; if it's at all questionable that a set of outcomes are equally likely, they aren't!
In summary: if you’re told to assume something, assume it; if it makes a big difference, check it; if it makes sense to expect it, expect it; and if you’re just in a class, relax and go with common sense.