00:01
Okay, in this problem, we're going to be working with probability and independent events.
00:06
So let's start off with a definition.
00:09
The definition, a and b are independent if the probability of a intersect b equals the probability of a times the probability of b.
00:30
Okay, so that's what it means for two events to be independent.
00:37
Let's look at the first part you want to show that if a and b are independent then so are a and b complement so i'm going to use a rule called the law of total probability and it says this the probability of an event a because the probability of a intersect b plus the probability of a intersect b if we're going to apply it to this example, we want to look at the probability of a intersect b complement.
01:31
That will be the probability of a minus the probability of a intersect b by rearranging the above formula.
01:49
And then because a and b are independent, we can write this as the probability of a times the probability of b and factor out the probability of a from both terms to get one minus the probability of b.
02:04
And one minus the probability of b is the same thing as the probability of b complement.
02:17
So that checks out, meaning that a and b complement are independent.
02:30
Okay.
02:33
The second part is very similar.
02:38
You want to work with a complement and b.
02:43
Can do that by doing the same method, the probability of a complement intersect b equals the probability of b times or minus the probability of a intersect b.
03:04
So i'm using b here because b is not changed on the left hand side.
03:10
That allows me to factor out the probability of b.
03:13
You have one minus the probability of a because this part here is the probability of a times the probability of b because a and b are independent.
03:27
This becomes the probability of b times the probability of a complement.
03:34
So once again, that checks out.
03:40
And the third part is pretty similar to.
03:46
You're looking at a complement and a b complement...