00:01
We know that a and b are independent.
00:03
Independent means they do not influence each other, which means the probability of a given b is equal to the probability of a.
00:12
Whether or not b has happened doesn't matter.
00:14
Probability of a is the same.
00:15
Same applies for b given a equals b.
00:19
And the probability of a and b is equal to probability of a times the probability of b.
00:25
I want to prove that a ' and b ' are also independent.
00:30
These are the complements.
00:32
So i'm going to add in some ingredients here, some formulas you should know.
00:37
The probability of x plus the probability of x ' is 1.
00:44
X plus its complement is 1.
00:47
The probability of x union y is equal to the probability of x plus the probability of y minus the intersection x and y.
00:58
This is the general addition formula.
01:00
Finally, the probability of x intersects y dash, while in here, is the probability of x prime union y prime.
01:15
So if i draw a quick venn diagram to display this one, if we have a and b, we have the intersection here, a and b, the complement to the intersection, so not the intersection, could be not b, it could be not a.
01:32
Between those i've got everything.
01:34
So if you take a prime union b prime, you get the complement to the intersection of a and b.
01:42
Now let's put these ingredients together.
01:45
So i know a and b is a times b.
01:50
And i'm going to take the complement to that.
01:53
I know this complement is 1 minus this value.
01:56
So let's put this as we've got a intersect b, and we prime that, is a prime union b prime.
02:14
Okay, 1 minus the probability of a and b from here is equal to the probability of a prime union b'.
02:27
I've just added probabilities in here.
02:29
Now let's apply the general addition formula to that right -hand side.
02:36
A', plus b', minus a ' and b'.
02:45
Okay, now i want to get some of these on their own.
02:53
Let me make some space...