Course Notes - Department of Mathematics and Statistics

PROBABILITY OF X SUCCESSESThe probability of x successses where x takes the values 0 to n isgiven by:( nP r (X = x) = πx)x (1 − π) n−xWhere ( nx) is the binomial coefficient( n n!=x)x! (n − x)!Consider the TuataraStephens Island is an uninhabited island in Cook Strait where tuataraare being re-established. For some years three locations havebeen visited on the island and tuatara have been found at a locationwith probability 0.4. At any visit X represents the number of locationsout of three at which tuatara are observed (X can take values0, 1, 2, or 3). Find the probabilities that 0, 1, 2 or 3 locations havetuatara on a visit. T is the event ‘location has tuatara’ and N is thecomplementary event ‘location has no tuatara’.Calculating Probabilities• Find the probability of seeing Tuatara at two of the three sites.P r(X = 2) = P r(T T N, T NT, NT T )= 0.096 + 0.096 + 0.096= 0.288• This is a binomial example with n = 3, π = 0.4.• In this case we are interested in the probability of two successes.64