A nice philosophic discussion of the use of statistics i in science from the Bayesian point of view can be found in Howson and Urbach (2006 ), which is available in paperback for modest cost . A more mathematical presentation is given by Jaynes (2003 ). Unions and Intersections of Probability - Venn Diagram The probability of some event E happening is written as P(E). The probability of E not happening must be 1 - P(E). The probability that either or both of two events E , and E, will occur is called the union of the two probabilities and is given by , P ( E, UE, ) = P(E,) + P(E2 ) _ P(E, nE2) (1.7 ) where P(E, E, ) is the probability that both events will occur , and is called the intersection . It is the overlap between the two probabilities and must be subtracted from the sum . This is easily seen using a Venn diagram , as below . In this diagram the area in the rectangle represents the total probability of one , and the area inside the two event circles indicates the probability of the two events . The intersection between them gets counted twice when you add the two areas and so must be subtracted to calculate the union of the probabilities . If the two events are mutually exclusive e , then no intersection occurs Fig . 1.2 Venn Diagram illustrating the intersection of two probabilities Another important concept is conditional probability . We write the probability that E, will occur , given that E, has occurred as the postulate P(E, | E] ) = - P (E, nE2 ) (1.8 ) P ( E , ) Changing this conditional probability relationship around a little yields a formula for the probability that both events will occur , called the multiplicative law of probability P(E] n E2) = P(E2 ( E] ) . P(E] ) = P(E] | E2 ) . P(E2 ) (1.9 ) If the two events are completely independent such that P(E] [ E, ) = P(E, ) , then we get