function [] = chap_3_prob_28()
% 
% Written by:
% -- 
% John L. Weatherwax                2006-08-28
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

format rat

% Part (a):
%

% for p(F):
%
% the first 19 cards are not aces:
firstPart = prod( 48 : -1 : ( 48-19+1 ) );
% the 20th card is an ace:
midPart = 4; 
% the remaining cards can be anything: 
lastPart = factorial( 52-20 );

num = firstPart * midPart * lastPart ;
den = factorial( 52 );

pF = num/den;
disp(pF)

% for P(EF):
%
% the first 19 cards are not aces:
firstPart = prod( 48 : -1 : ( 48-19+1 ) );
% the 20th card is an ace (but not the ace of spades):
midPart = 3; 
% the 21st card is the ace of spades:
midPart2 = 1;
% the remaining cards can be anything: 
lastPart = factorial( 52-21 );

num = firstPart * midPart * midPart2 * lastPart ;
den = factorial( 52 );

pEF = num/den;
disp(pEF)

% the probability we want:
%
pEF/pF


% Part (b):
%

% P(EF) is now some what different:
%
% the first 19 cards are not one of the four aces or the two of clubs
firstPart = prod( 47 : -1 : ( 47-19+1 ) );
% the 20th card is any ace:
midPart = 4;
% the 21st card is the two of clubs:
midPart2 = 1;
% the remaining cards can be anything: 
lastPart = factorial( 52-21 );

num = firstPart * midPart * midPart2 * lastPart ;
den = factorial( 52 );

pEF = num/den;
disp(pEF)

% using this the probability we want is given by 
%
pEF/pF



% the second method:
% 
pA1CA2C = prod( 30:48 ) / prod( 34:52 )
pF = pA1CA2C * ( 4 / 33 ) 
pEF = (1/32) * (3/33) * pA1CA2C