# Useful Formulae

Given two numbers,aand b,then the three important averages or means,traditionally used in geometry,music and architecture,are the arithmetic mean, ab, the geometric mean, ab and the harmonic mean, Suppose we have a situation with npossible, equally likely outcomes,kof these being desired. The probability pof a desired outcome occurring on any one occasion is then p Note that pis always between 0 and 1. ImagineEand Fare two independent possible events, with probabilities PE and PF of occurring,respectively. The probability of both Eand Foccurring is PEF PExPF.With this,we can express the probability of eitherEorFoccurring asPEF PEPFPEF. Fis not independent of Ewhen the probability of Foccurring is altered by Eoccurring. In this case we define the conditional probability PFEto be the probability of Foccurring once Ehas occurred,and we have PEF PExPFE. For example,if G and H both represent the choice of a black ball from the bag on our left,then PGH x see back page for fractions. 20 MEANSANDPROBABILITIES safe proportions and risky outcomes number of desired outcomes total number of possible outcomes ab 2ab 5 2 41 10 1 n k 21