 There's many questions mathematicians ask whenever encountering a new concept, but sooner or later they do get around to asking the question, how do we calculate the values of this new concept? So let's consider. Suppose our random experiment was performed K times. We'll do a little hand-waving and use our frequentist interpretation of probability. Then the probability of some event B would be the number of times B occurred over K. We could also find the probability of A and B occurring simultaneously, which would be the number of times both A and B occurred over K. And let's take a look at the conditional probability, the probability that A occurred given that we know B occurred. And this is a little bit more complicated. Since we know that B occurred, we can focus on the number of times that B occurred. And of those times, we're interested in the number of times both A and B occurred. Now if you look at these expressions carefully, you see that if we divide the probability of A and B by the probability of B, the divisor of K is going to drop out and we'll get this probability of A given B. And this suggests the following. We can define the conditional probability of A given B as the quotient of the probability of both A and B and the probability of B. So for example, suppose we have a number of red and green tokens falling into a field. And if we look at where these tokens fall, we might consider the following probabilities. The probability that a token is green given that it is above the horizontal line, the probability that a green token is above the horizontal line. Now these are probabilities we could have found before, but now we have this nice computing formula. And so let's show that the result that we get is consistent with the definition given for computing conditional probability. So let's take a look at this carefully. Let G be the event that a token is green and H the event that a token is above the horizontal line. In the first case, we know that the token is above the horizontal line and we want to find the probability it is green. So if we look, we find there are 17 tokens above the horizontal line and of these, 8 are green. And so that says that the probability that a token is green given that it's above the horizontal line is 8 out of 17. In the second case, we know that the token is green and we want to find the probability it is above the horizontal line. So again, we take a close look at our picture and we find there are 18 green tokens and of these, 8 are above the horizontal line. So the probability a token is above the horizontal line given that it's green is 8 out of 18. So we'll pull in our formula for calculating conditional probability to find the probability of G given H we want to find the probability of G and H divided by the probability of H. Likewise, the probability of H given G is the probability of G and H over the probability of G. To use the formula, we need to find probability G and H the probability a token is green and above the horizontal line. It's important to recognize that this is not a conditional probability and so we know nothing about the token. So we find stuff out and in this case we find there are 29 tokens altogether and there are 8 tokens that are green and above the horizontal line. So the probability a token is green and above the horizontal line is 8 out of 29. We also need the probability of H the probability a token is above the horizontal line and the probability of G the probability a token is green. So we find there are 17 tokens above the horizontal line out of 29 tokens altogether so the probability of H is 17 out of 29. There are 18 green tokens so the probability a token is green is 18 out of 29 filling those into our conditional probability formulas and simplifying gives us the probability of G given H is 8 out of 17 which checks and likewise the probability of H given G is 8 out of 18 which also checks. The value of these computing formulas is it allows us to talk about more probabilities so let's consider independence again. Suppose a and b are independent events that means that a did not cause b and b did not cause a or does it? If you only learn one thing about probability you're going to fail the class but among the very important things you should learn is that independence has nothing to do with causality and remember definitions are the whole of mathematics all else is commentary since we want to talk about independent events let's bring in that definition that means that the probability of a given b is equal to the probability of a itself so if a and b are independent events then we know that the probability of a is the same as the probability of a given b now the computing formula for conditional probability tells us that the probability of a given b is equal to this quotient and equals means replaceable wherever I see the one side I can replace it with the other side so I see probability of a given b I can replace it with the quotient and I'll do a little algebra I'll multiply by probability of b so I can get rid of the fraction and this gives me a useful relationship suppose a and b are independent events then the probability of a and b both occurring is the product of the probability of a with the probability of b and this gives us a fantastically useful formula as long as we read the fine print a and b have to be independent events so here's an important question how can we determine if a and b are independent events as a general rule the only way to determine if events are independent is to use your knowledge of life the universe and everything what's important to remember independence has nothing to do with causality instead, what's important to assess is whether our knowledge that b has occurred changes our confidence that the event a has occurred for example, suppose two six-sided dice are ruled what's the probability that the first die shows a 5 while the second die shows a 3 so we might consider this, we have two events a, the first die shows a 5 and b, the second die shows a 3 so the probability of a that the first die shows a 5 well, there's six possible outcomes and they seem to be equally likely so we'll reluctantly conclude that that probability is going to be one sixth now if we want to use our formula the fine print says that the events have to be independent so let's find the probability of a given b and here's where that Bayesian viewpoint is helpful given that you know that the second die shows a 3 does this change your expectation that the first die will show a 5 and it seems like this extra information doesn't cause us to revise our probability so it seems that a and b are independent events so the event that the first die shows a 5 and the second die shows a 3 is the event a and b and since a and b are independent or at least we hope so the probability of a and b is the probability of a times the probability of b one sixth times one sixth or one thirty sixth