%% Discrete probablility simulations, Grinstead and Snell section 1.1
% *Nancy K. Stanton*
% 
% code based in part on Peter Doyle's code
%% Random numbers in MATLAB
% MATLAB has the command *rand* to generate random numbers between 0 and 1.
rand
%%
% *rand(1,n)* will produce an $n$ component vector of random numbers, so to
% get 5 random numbers, I could give the command
rand(1,5)
%%
% To get a table of 20 random numbers like the one on p. 3 which has 5 rows
% of 4 numbers, I can give the command  
rand(5,4)
%% Simulating cointossing
% I can simulate tossing a fair coin by generating a random number and
% calling it heads if it is less than 1/2 and tails if it is greater.
% The MATLAB function *cointosses.m* simulates this. To see how it works, I
% can give the command  *doc cointosses* which will pop up the help for 
% the command in a separate window.
doc cointosses
%%
% So, if I want to toss a fair coin 20 times, I give the command
cointosses(20,1/2)
%% 
% To see the code, I can give the command
type cointosses
%%
% Let's see what happens if we toss the coin 1000 times and repeat this a
% 5 times. We probably don't want to list all 1000 outcomes, just the
% proportion, which we can do with *cointosses2* which is the same as
% *cointosses* except that it only displays the proportion of heads.
cointosses2(1000,1/2)
%%
cointosses2(1000,1/2)
%%
cointosses2(1000,1/2)
%%
cointosses2(1000,1/2)
%%
cointosses2(1000,1/2)
%% de Mere's first experiment
% Is it reasonable to think that in 4 rolls of a die a 6 will turn up? 
% The command *demere1(n)* simulates this experiment n times.
demere1(100)
%%
demere1(1000)
%%
demere1(1000)
%% 
demere1(1000)
%%
demere1(1000)
%% de Mere's second experiment
% We'll try this 5 times with 10000 trials, 24 rolls per trial.
demere2(10000,24)
%%
demere2(10000,24)
%%
demere2(10000,24)
%% 
demere2(10000,24)
%%
demere2(10000,24)
%%
% Now we'll do it with 25 rolls, which is the default for *demere2*.
demere2(10000)
%%
demere2(10000)
%%
demere2(10000)
%% 
demere2(10000)
%%
demere2(10000)
%%
% The commands *cointosses*, *cointosses2*, *demere1* and *demere2* are not
% built-in MATLAB commands, so if you want to use them, you'll have to
% download the m-files  and put them in your MATLAB path.