Could someone explain to me the Rule of Compound Probability involving "and"?

independent ##p(a,b) =p(a)*p(b)## dependent ##p(a,b) =p(a|b)*p(b)##

The probability of two events can be thought of happening in two ways. 1) When knowing something about one tells you nothing about the other or independence. For example what is the probability of raining and the probability of flipping a coin and landing heads.

##p(a,b) =p(a)*p(b)##

2) When knowing something about affects the other or dependent. For example what is the probability of boys in a high school math class and the probability of seniors. Since its possible to be a boy and a senior we will likely double count these outcomes thus the rule is

##p(a,b) =p(a|b)*p(b)##