Introduction | Probability | Achievable AMC 10/12

Probability (and combinatorics)

What’s your gut reaction when you see the words “how many” in a question? How about “what is the probability”?

Let’s make you a master of both probability and combinatorics so that you can not only get to AIME, but also do well once you’re there. But first, a little refresher:

The basics

Definitions

Probability: The study of how likely things are to happen.
Combinatorics: The study of counting up how many ways things can be put together. It is often used to count the different events that could happen, especially when you can think of “events” as what you get when you put together some conditions.

Combinatorics and probability are usually discussed together. Sometimes we even use the words interchangeably. That’s because calculating the chance something will happen (probability) and counting up the different ways something could happen (combinatorics) are very closely related.

And that is because the way to calculate the probability of something happening the way you want it to is to do this: you count up the number of ways you could get what you want, then you count up the number of ways you could get anything at all, and then you divide the first total by the second. Assuming all the different events were equally likely, this’ll work fine.

So, for example, if you want to know the chance of rolling a 5 or higher on a regular die, you’d count up that there are two ways to get what you want (you could get a 5 or you could get a 6), and you’d count up that there are six ways to get anything at all (you could get anything from a one to a six). So there’s a two in six (one in three) chance of getting at least a five when you roll one die.

Since combinatorics is the branch of math that covers counting up the different ways something can happen, it’s really really useful for calculating probabilities. And it turns out that calculating probabilities – i.e. predicting the future accurately – is a super-valuable thing to do with math.

So of course you want to get good at it, and --lucky you-- the AMC tests it.

Now, even if you get nothing else out of this, I want you to get these two core ideas:

Calculating probability is easy as long as you’re careful to keep track of everything correctly.
Calculating probability is mostly about counting.

But if you want to get more out of this, then you need to get really good at the basic operations that lead to being able to count big sets both quickly and accurately.

That means that you’ll need to be able to make all the following claims with justified confidence:

Pop quiz

True or false:

I can write out any row of Pascal’s Triangle without stopping to decide what to write
I can explain what “factorial” means
I can do relatively complicated-looking algebraic manipulation on factorials without a ton of trial and error
I know what $(k n)$ , also known as “n choose k”, means
I know some values for $(k n)$ offhand; for example, I know that $(2 4)$ is $6$
I know how to represent the difference between “Event A and Event B both happen” and “Event A or Event B happens” both algebraically and on a Venn diagram

Examples

Write out the fourth row of Pascal’s Triangle.

(spoiler)

The fourth row of Pascal’s Triangle is $1, 4, 6, 4, 1$ .

Write out the first three terms of the $n$ th row of Pascal’s Triangle.

(spoiler)

The $n$ th row of Pascal’s Triangle starts with: $(0 n)$ , $(1 n)$ , $(2 n)$ .

Simplify $(k n) + (k - 1 n)$ .

(spoiler)

Ooh, tricky. Even if you’ve memorized some formulas in this area, you probably haven’t memorized one that fits this problem exactly.

But you might know that $(k n) = (n - k n) = (k n - 1) + (k - 1 n - 1)$

And you might also realize that this applies even to unusual presentations of $n$ (such as $n + 1$ , as in this case).

That means that $(k n) + (k - 1 n) = (k n + 1)$ , which is a lot tidier than we might expect at first.

…

Side note: by definition, you could have come up with the answer $\frac{n !}{( n - k )! k !} + \frac{n !}{( n - ( k - 1 ))! ( k - 1 )!}$ instead. That is equivalent to the expression in the question, but it’s a wrong answer nonetheless, because the instructions say to simplify. (And also because your first reaction to seeing something complicated should be reflection rather than brute force.)

What does $P (A \cup B)$ mean, and how is it related to $P (A)$ and $P (B)$ ?

(spoiler)

$P (A \cup B)$ is defined as the probability that event $A$ or event $B$ or both will happen.

But if you consider the Venn diagram of $A$ and $B$ here you will see that $P (A) + P (B)$ double-counts the overlap area. Therefore we must subtract it once in order to get the correct total probability:

$P (A \cup B) = P (A) + P (B) - P (A \cap B)$

Special note about the Probability units in this course

The other Probability units of this course don’t have new quizzes in them. The probability quizzes are all in this unit instead. There are two reasons for this:

Most of the memorizable probability stuff is contained in the Combinatorics unit. I think that this makes for a more sensible organization overall.
Most students either know this material very well, or not well at all, with few in between.

So if you are in the “expert” camp already, then you have followed the instructions above and added the Probability quizzes to your deck; you will next read through the rest of the Probability unit, then move on to other topics.

If you are not yet in the “expert” camp, then I have two recommendations for you:

Read through the rest of the Probability units in this course. That will give you a good basic grounding in how this stuff works. That’s especially important, because although a lot of this stuff seems obvious, there’s also a lot here that contradicts most people’s intuition. It will just feel wrong at first. So doing some extra reading about it and thinking about it before you start problems will help a lot.
Reading through Chapters 25, 26, and 27 of AoPS Volume 1, and doing a few of the many problems there, is one of many good ways to reinforce the basic ideas you need in order to get full value out of this unit of the course. If you don’t have that book, I recommend buying it. (Note: I’m not related to AoPS in any way, and they don’t pay me anything to say this. I just think it’s a good book.) If you’d rather not, though, don’t worry; a search will lead you to many other great resources as well.