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Fundamental counting principle (probability)

Fundamental counting principle (probability)

Author: Christopher Danielson

To demonstrate the fundamental counting principle, for use in describing sample spaces composed of equally likely outcomes.

The Fundamental Counting Principle is typically discussed in the context of combinatorics, and the situations are often contrived. This packet seeks to illustrate the principle in the context where it is most useful for most people-that of probability.

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The Fundamental Counting Principle states,

If two choices are to be made, with A ways to make the first choice and B ways to make the second choice, then there are A x B ways to make the two choices.

And the principle generalizes to three or more choices (A x B x C, etc.)

The standard examples and applications of the Fundamental Counting Principle at the introductory level include the following:

  1. You have three pairs of pants (blue, black and khaki) and four shirts (white, black, gray and taupe). How many outfits can you make?
  2. The school cafeteria offers three main dishes (slop, glop and mush) and three types of milk (skim, whole and chocolate). How many different lunches can you make?
  3. A state's license plates have three letters, followed by three numbers. How many different plates are possible?

In each of these cases, the Fundamental Counting Principle applies. But does anyone really look at their wardrobe this way? And is slop with whole milk really all that different from slop with skim? In a sense, while the principle applies, the problems are not very convincing.

The only one of these that seems plausible is the license plates. And even then, it's not all that compelling a problem. And it ignores the fact that certain of these plates ought not be allowed on the road.

Where really need the Fundamental Counting Principle in high school and early college mathematics is in the study of probability. In particular, when we want to dig down deep into a sample space, we often need to use the Fundamental Counting Principle to make sure we have identified all possible outcomes.

Sample space

A Sample Space is the set of all possible outcomes in a probability situation. The idea of a sample space is so important that it has its own symbol, the Greek capital letter omega:

Exactly how you write down your sample space depends quite a bit on what you are using it for. Consider the simple case of rolling two six-sided dice. Some different purposes and their associated sample spaces are these...

If you are playing Monopoly and you need doubles to get out of jail:

If you are three spaces away from Free Parking and hope to land there on this turn:

If you are playing a game in which you get 1 point for rolling an even product:

One of the biggest mistakes in all of probability is to assume that the elements of your sample space are equally likely. Finding theoretical probabilities is much easier if we have a sample space composed of equally likely elements, so this is a desirable thing to have, but it does not follow that listing out the sample space means the elements are equally likely.

If we wrongly assume that each sample space above is composed of equally likely elements, then we conclude that the probability of rolling doubles is 1/2, the probability of rolling a 3 is 1/2 and the probability of rolling an even product is 1/2. But none of these turns out to be correct.

So the thing to do is to dig deeper and write out the sample space in more detail. Consider the even/odd product scenario above. Rolling an even number is an event composed of several outcomes. I might get a product of 2, or 4. Both of these are even. So a more detailed sample space for this scenario is the following:

These are all of the possible products. But we again have to ask whether the elements of this sample space are equally likely. It turns out that they are not. There is only one way to get a product of 1: rolling double 1's. But there are at least two ways to get a 4: rolling a 1 and a 4, and rolling double 2's. So just as in the first examples, we cannot go from the sample space directly to computing probabilities.

It turns out to be possible in this case to list out the sample space in such a way that all of the outcomes are equally likely. Doing so, and in particular counting the number of elements when we do so is a realistic use of the fundamental counting principle.

Fundamental counting principle used for sample spaces

Recall the statement of the Fundamental Counting Principle:

If two choices are to be made, with A ways to make the first choice and B ways to make the second choice, then there are A x B ways to make the two choices.

When we roll two dice, there are (in a sense) two choices getting made. The first die has to "choose" a number, and the second die has to "choose" a number. Since these are normal six-sided dice, we know that A is 6, B is 6 and that each of the 6 choices on each die is equally likely. So, by the Fundamental Counting Principle, we know that there are 6x6=36 total choices, i.e. that the sample space consists of 36 equally likely outcomes. Namely:

But how many of these give even products?

Recall from elementary school arithmetic that:

We want to count the number of ways to get an even product. We can use the Fundamental Counting Principle here. We'll do it three times-once for each of the ways to get an even product listed above.

odd x even = even

There are three equally likely choices for an odd number on the first die: 1, 3 and 5. There are three equally likely choices for an even number on the second die: 2, 4 and 6. So A is 3, B is 3 and the Fundamental Counting Principle gives 3x3=9 ways to get an even number this way.

even x even = even

Here too, there are three equally likely choices for an even number on the first die: 2, 4 and 6. Same for the second die, so A is 3, B is 3 and we have 3x3=9 ways to get an even number this way.

even x odd = even

A=3, B=3, so 3x3=9 ways here too.

All together

We conclude that there are 9+9+9=27 equally likely ways to roll an even product. Now we can use the definition of theoretical probability:

The Fundamental Counting Principle is behind most of the ways we count the number of elements of a sample space, including permutations and combinations.