/* 

  Four cards in Picat.

  From Bar-Hillel & Falk "Some teasers concerning conditional probabilities" 
  """
  Problem 3:
  A deck of four cards consists of the ace of spades, the ace of clubs, the deuce of spades, 
  and the deuce of clubs. A hand of two cards is randomly dealt from this deck. What 
  is the probability that it contains both aces if we know it contains at least one?

  The answer to this problem is traditionally agreed upon to be l/S, following the reasoning 
  that five equiprobable hands are compatible with the conditioning event (only the double 
  deuce hand is ruled outj, and just one of these contains both aces.

  Now compare Problem 3 to the following: 

  Problem 4:
  Like Problem 3, but the question is: What is the probability that the hand contains 
  both aces if we know it contains the ace of spades’.
  """

  Here are three runs:
  * none  : No observation
  * model1: At least one ace
  * model2: It contains ace of spades

  Cf 
  - ppl_two_children_problem.pi
  - ppl_how_many_sons.pi
  - my Gamble model gamble_four_cards.rkt

  This program was created by Hakan Kjellerstrand, hakank@gmail.com
  See also my Picat page: http://www.hakank.org/picat/

*/

import ppl_distributions, ppl_utils.
import util.
% import ordset.

main => go.

/*
  obs = none
  var : selected
  Probabilities:
  [AC,2S]: 0.0886000000000000
  [AC,2C]: 0.0857000000000000
  [AS,AC]: 0.0856000000000000
  [AC,AS]: 0.0848000000000000
  [2C,AC]: 0.0848000000000000
  [2C,AS]: 0.0843000000000000
  [2C,2S]: 0.0836000000000000
  [2S,AC]: 0.0832000000000000
  [AS,2S]: 0.0818000000000000
  [AS,2C]: 0.0804000000000000
  [2S,AS]: 0.0789000000000000
  [2S,2C]: 0.0783000000000000

  var : num aces
  Probabilities:
  1: 0.6677000000000000
  2: 0.1704000000000000
  0: 0.1619000000000000
  mean = 1.0085

  var : p
  Probabilities:
  false: 0.8296000000000000
  true: 0.1704000000000000
  mean = [false = 0.8296,true = 0.1704]


  obs = model1
  var : selected
  Probabilities:
  [AC,2S]: 0.1082604004316029
  [AC,2C]: 0.1058626064021101
  [2C,AC]: 0.1035847020740918
  [AS,AC]: 0.1026255844622947
  [AC,AS]: 0.0997482316269032
  [2C,AS]: 0.0977101067018343
  [AS,2S]: 0.0972305478959357
  [AS,2C]: 0.0961515405826640
  [2S,AC]: 0.0953123126723414
  [2S,AS]: 0.0935139671502218

  var : num aces
  Probabilities:
  1: 0.7976261839108021
  2: 0.2023738160891979
  mean = 1.20237

  var : p
  Probabilities:
  false: 0.7976261839108021
  true: 0.2023738160891979
  mean = [false = 0.797626,true = 0.202374]


  obs = model2
  var : selected
  Probabilities:
  [2S,AS]: 0.1723726114649682
  [AC,AS]: 0.1709792993630573
  [AS,AC]: 0.1681926751592357
  [AS,2S]: 0.1675955414012739
  [AS,2C]: 0.1648089171974522
  [2C,AS]: 0.1560509554140127

  var : num aces
  Probabilities:
  1: 0.6608280254777070
  2: 0.3391719745222930
  mean = 1.33917

  var : p
  Probabilities:
  false: 0.6608280254777070
  true: 0.3391719745222930
  mean = [false = 0.660828,true = 0.339172]

*/
go ?=>
  member(Obs, [none,model1,model2]),
  println(obs=Obs),
  reset_store,
  run_model(10_000,$model(Obs),[show_probs,mean]),
  nl,
  % show_store_lengths,nl,
  fail,
  nl.
go => true.
  
  
model(Obs) =>
  Cards = ["AS","AC","2S","2C"],
  Selected = draw_without_replacement(2,Cards),
  NumAces = [ 1 : S in Selected, membchk(S,["AS","AC"])].sum,

  if Obs != none then
    if Obs == model1 then
      % We observe at least one ace
      observe(NumAces >= 1)
    else
      % We observe the Ace of Spades
      % observe(membchk("AS",Selected)), % Picat complains about this
      observe(cond(membchk("AS",Selected),true,false)),
    end
  end,

  % What is the probability of two aces?
  P = check(NumAces == 2),

  if Obs == none ; observed_ok then
    add("selected",Selected),
    add("num aces",NumAces),
    add("p",P)
  end.