
clear all
close all

  disp('CLT in Action: AVIO Company')
  lw = 2; 
  set(0, 'DefaultAxesFontSize', 16);
  fs = 15;
  msize = 10;
  

% Avio Company.============================================
%Avio company sells 410 plane tickets for the flight at which physically
%only 400 seats are available. Find the probability that company
%overbooked the flight if a person who buys a ticket shows at the gate 
%with probability of 0.96.

% the mean
 meavio =410*0.96
% expected number of shows is 393.6000

% the std
stdavio = sqrt(410*0.96*0.04)
%theoretical std is 3.9679

 1-normcdf((400.5-meavio)/stdavio)  % 0.0410
%Notice that in this case the approximation is not very good
%since the exact probability is
exact = 1-binocdf(400, 410, 0.96)
%ans = 0.0329, which is big relative error (not absolute)

%It is interesting that the simulation gives better
%answer than the normal approximation
simu = sum(binornd(410, 0.96, 10000,1) > 400)/10000
%simu = 0.0313
%Better than the normal is the Poisson approximation;
% the probability of overbooking is 
% equal to probability of 9 or less noshows:
poisscdf(9, 0.04*410) %prob that 9 or less fail to show
%ans =  0.0355




