We might also talk about the complement, or

what's the probability of A not occurring.

Well, the notation for A not occurring would be a probability of A with

a superscript c, can be calculated as one minus the probability of A.

So if we think about this as what's the probability that somebody submits

a complaint versus not submitting a complaint after bad customer service?

What's the probability of a product being successful in one launch

versus the probability of it being a failure?

These are examples where they're the complements of each other.

Either you choose to submit a complaint or you do not.

Either the product is a success or it's deemed a failure.

Well since they're the flip side of each other and the probabilities have to add up

to one, that's where we get one minus the probability being the complement.

Just to give a visual representation of this,

one of the rules of probability is the probability of A or B occurring.

And A or B, the notation for that, is the union of A and

B or A then what looks like a U.

And so how can this happen?

Either event A can occur, or event B can occur.

And that's what these first two pieces are indicated for us in this equation.

But then if you think about the likelihood of A occurring.

The likelihood of B occurring.

Sometimes when A happens, B also happens.

So, we're also double counting the probability

of both of these events happening together.

And so we actually have to subtract out the joint probability.

So what this looks like visually is I've got the probability of A,

I've got the probability of B.

And the overlap between those circles, that's the probability of A and B.

So what we want to calculate is how likely it is that A occurs?

So that's the first sphere.

How likely B occurs, so that's our second sphere.

And that's the probability of A plus the probability B.

But what we've ended up doing is double counting this shaded region,

which is the joint probability of A and B.

So that's why we've got to subtract that out.

And then what we're left with is.

The overall shaded area that's the probability of A or B occurring.

All right so here just a couple of examples.

You buy a particular brand at least once on your last two shopping trips.

So either you bought it on the first trip or you bought it on the second trip.

You also could have bought it, on both of those trips.

You make a late, another example, you're late paying at least one of your bills.

So, could be late on your mortgage, could be late on your car loan.

Could be late on your student loans.

Well, being late on any one of them.

So we're just, all that we're interested is that you're late on at least one bill.