# How to Manipulate among Joint, Conditional and Marginal Probabilities

### The equation below is a means to manipulate among joint, conditional and marginal probabilities.  As you can see in the equation, the conditional probability of A given B is equal to the joint probability of A and B divided by the marginal of B.  Let’s use our card example to illustrate.  We know that the conditional probability of a four, given a red card equals 2/26 or 1/13.  This should be equivalent to the joint probability of a red and four (2/52 or 1/26) divided by the marginal P(red) = 1/2.  And low and behold, it works!  As 1/13 = 1/26 divided by 1/2.  For the diagnostic exam, you should be able to manipulate among joint, marginal and conditional probabilities. # Bayes’ Theorem

### Bayes’ theorem: an equation that allows us to manipulate conditional probabilities. For two events, A and B, Bayes’ theorem lets us to go from p(B|A) to p(A|B) if we know themarginal probabilities of the outcomes of A and the probability of B, given the outcomes of A. Here is the equation for Bayes’ theorem for two events with two possible outcome (A and not A). # Bayes’ Theorem Example

### We will call p(cancer) = P(A), and the P(positive test) = P(B). We want to know P(A|B)–the probability of having cancer if you have a positive test. 