Here’s the way to calculate the probability of something happening the way you want it to: 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. 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.
There are two main things I want you to get out of this:
Combinatorics is the branch of math that covers counting up the different ways something can happen. So 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 8 tests it.
Now, earlier I said you have to keep track of everything carefully. That’s because advanced probability questions aren’t hard; they’re merely complicated. So let’s cover some of the complications, so that you can keep everything straight when they try to confuse you.
The main way these things get complicated is that they give you multiple details, and you have to not only keep track of them, but also know how to combine them into one answer by the end. So know these rules:
If you want to know the chance of having two things both happen, you calculate their chances individually, then multiply the probabilities together. So, for example, if there’s a 1/3 chance that it will rain on Tuesday, and a 1/3 chance that it will rain on Wednesday, then there is a 1/9 chance (1/3 * 1/3) that it will rain on both days.
We often shorthand this by saying “and means times.” I.e. the chance that it rains Tuesday and rains Wednesday is 1/3 times 1/3.
And means times.
On the other hand, “or means plus.” If you want to know the chance that one thing happens or the other happens, you calculate the probabilities then add. For example, if there’s a 20% chance it rains Friday, then there’s an 80% chance it won’t. Therefore the chance that it will either rain or not on Friday is 20% plus 80%. And that makes sense: of course there is a 100% chance that on Friday either it will rain or it won’t.
Or means plus.
If you only remember those two rules, you will be able to get at least half of the combinatorics and probability problems right.
So then what about the other half?
Well, unfortunately, the rules above only apply in certain (common) circumstances. Since the circumstances are common, the rules work most of the time. But if you want to know what to do all of the time, you have to know about independent events.