It’s not difficult. It’s merely complicated.

In the last unit you got the basics of probability (and, let’s be honest, some expert tools as well).

But surely if the “basics” are that hard, we should run screaming from the tough problems in this area, right?

Not so fast. Advanced probability questions routinely seem crazy hard at first, and that scares most people off. But what if I showed you that they can actually be easy problems that give you easy points?

It all comes down to the fact that advanced probability questions usually 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. And remember what I said in the last unit: you just have to keep track of everything carefully. That goes double for this unit!

The rules

“And” means times.
“Or” means plus.
The above two rules are only true sometimes.

I know, that last rule is rough! I want to explain, but it’s complicated. For now, let’s go with a simplification that works just fine most of the time:

If two events are unrelated, then you can use “and means times”.
If two events are related, then you use “or means plus”.

We’ll get into more detail on this in the next unit, but for now I’ll restrict my examples to ones where the first two rules do apply.

A bit more detail

The main way these things get complicated is that they give you a lot of information, and you have to not only keep track of it all, but also know how to combine it all into one answer by the end. Let’s cover the first two rules, and how they allow you to combine correctly:

And means times.

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.

Or means plus.

Another shorthand, meaning that if you want to know the chance that one thing happens or the other happens, you calculate the probabilities, then add.

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.

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 first two 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, covered in the next unit.

What to do

Like I mentioned earlier, there aren’t any quizzes for this unit. But here’s something really useful you could do instead: I recommend that you go back to a probability question that you had trouble with in the past and try to solve it again, only this time using at least three full sheets of paper, in combination with the rules above. I predict you’ll discover some paths that you missed the first time. And maybe one of them will even turn out to be useful.