IEE 380 Ch 2 Total Probability Rule

well now we're going to talk about something called conditional probability and we already know that the probability of a is this notation and let's say we have another event D you call E1 E2 it doesn't matter we already know that the probability that event A and B occurring together would be this intersection

symbol well there's a new symbol we're going to learn here it's just a a vertical line and we would read this as the probability of a given B so what it does is it changes our sample space if we're conditioning on a particular event and uh if they're dependent on each other so

here is the introduction to something called the total probability rule which is going to use both of these notations here okay so here's my diagram suppose that this is also called sometimes we call this the law of total probability I use the terms interchangeably now what if I'm going to grab some colored

pens here I've got a yellow perfect so suppose that I have these three events E1 E2 and E3 and the whole sample space is this rectangle so s is going to be the whole rectangle E1 is going to be I'm going to use different colors here this green area here so I'm just going to kind of

shade this it's a little bit irregular it's kind of bordered by this wave looking thing here so that's going to be I call that E1 e 2 I'm going to call the piece that has this borderline here so this is going to be E2 and then E3 is going to be the remainder

of that rectangle I'm going to get a color that contrasts nicely there let me use orange here so this will be E3 so that's going to be all of the rest of it so if I add up that green area the black area and the orange area that equals s and the probability of s