IEE 380 Ch 2 Bayes Theorem

well we have learned about conditional probabilities so that I can teach you something called Bayes theorem um kind of interesting uh he was actually a a mathematician and a minister so what base theorem says and it's uh pretty well Quantified here in this formula pardon all the Shadows here let me get my pens over here

what base theorem says is that we can look at the conditional probability of a given that an event B has occurred by looking at kind of the opposite of that be given a times the probability of a over the probability of B and you may recognize this numerator actually as well we were talking about the law of

total probability this is just the probability of a intersected with B over the probability of B and you can switch this around too so it follows then that the probability of B given a just by doing a little bit of algebra here um that's just going to be the probability of a given B times the

probability of B over the probability of a and so we often need to use whoops the law of total probability to get this denominator here when we're looking for you know just the probability that a particular event has occurred so an example an example one here that I'm about to show I I show it only

because I want to demonstrate that Bayes theorem actually does actually does work to stick ignore my coffee stains over here so for example and I think I need some um I need some colored markers here so that's what I'm going to do some other ones so suppose that I take a six-sided die

I'm going to roll it twice I'm going to assume that those rolls are independent okay so but we're going to assume and they they are but um so the rolls let's write that down rolls are independent and the sample space for the rolls um I've got I'm gonna just use ordered pairs here so for example a four and a