/* 

  Brain Twister #44: Dice and cards in Picat.

  https://enigmaticcode.wordpress.com/2024/11/01/braintwister-44-dice-and-cards/
  """
  From New Scientist #3515, 2nd November 2024
  
  On each turn of a game, we roll three standard dice with faces numbered 1 to 6 and 
  add up the numbers shown.
  (a) What is the average (mean) value of that total over a long game?

  Now suppose instead of rolling dice, we draw three cards from a six-card deck 
  where the cards are numbered 1 to 6.
  (b) If we put each card back after drawing it and shuffle before drawing another card, 
      does this change the expected value?

  (c) If we instead draw three cards without replacement (so the probabilities are no longer 
      independent), what is the average value of the total?
  """


  Picat> X = sum_prob_dist_mean(1,6,3)                          
  X = 10.5

  All probabilities:
  Picat> pdf_all($sum_prob_dist(1,6,3),0.001,0.9999999).sort_down(2).printf_list 
  11 0.125 
  10 0.125 
  12 0.115740740740741 
   9 0.115740740740741 
  13 0.097222222222222 
   8 0.097222222222222 
  14 0.069444444444444 
   7 0.069444444444444 
  15 0.046296296296296 
   6 0.046296296296296 
  16 0.027777777777778 
   5 0.027777777777778 
  17 0.013888888888889 
   4 0.013888888888889 
  18 0.00462962962963 
   3 0.00462962962963 


  Cf 
  - ppl_sum_prob_dist.pi
  - my Gamble model gamble_brain_twister_44_dice_and_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.

/*
  """
  On each turn of a game, we roll three standard dice with faces numbered 1 to 6 and 
  add up the numbers shown.
  (a) What is the average (mean) value of that total over a long game?
  """

  var : s
  Probabilities:
  10: 0.1239000000000000
  11: 0.1218000000000000
  9: 0.1195000000000000
  12: 0.1162000000000000
  8: 0.0979000000000000
  13: 0.0947000000000000
  14: 0.0703000000000000
  7: 0.0686000000000000
  6: 0.0481000000000000
  15: 0.0457000000000000
  16: 0.0292000000000000
  5: 0.0275000000000000
  17: 0.0157000000000000
  4: 0.0129000000000000
  18: 0.0040000000000000
  3: 0.0040000000000000
  mean = 10.5087

*/
go ?=>
  reset_store,
  run_model(10_000,$model,[show_probs,mean % ,
                           % show_percentiles,
                           % show_hpd_intervals,hpd_intervals=[0.94],
                           % show_histogram,
                           % min_accepted_samples=1000,show_accepted_samples=true
                          ]),
  nl,
  % show_store_lengths,nl,
  % fail,
  nl.
go => true.
  
  
model() =>
  N = 3,

  X = dice_n(6,N),
  S = X.sum,
  
  add("s",S).


/*

  """
  Now suppose instead of rolling dice, we draw three cards from a six-card deck 
  where the cards are numbered 1 to 6.
  (b) If we put each card back after drawing it and shuffle before drawing another card, 
      does this change the expected value?
  """

  No, the estimated value is the same as in a).

  var : s
  Probabilities:
  11: 0.1232000000000000
  10: 0.1217000000000000
  12: 0.1194000000000000
  9: 0.1172000000000000
  8: 0.0964000000000000
  13: 0.0953000000000000
  14: 0.0733000000000000
  7: 0.0679000000000000
  6: 0.0478000000000000
  15: 0.0437000000000000
  5: 0.0267000000000000
  16: 0.0266000000000000
  17: 0.0167000000000000
  4: 0.0146000000000000
  18: 0.0051000000000000
  3: 0.0044000000000000
  mean = 10.5201

*/
go2 ?=>
  reset_store,
  run_model(10_000,$model2,[show_probs,mean % ,
                           % show_percentiles,
                           % show_hpd_intervals,hpd_intervals=[0.94],
                           % show_histogram,
                           % min_accepted_samples=1000,show_accepted_samples=true
                          ]),
  nl,
  % show_store_lengths,nl,
  % fail,
  nl.
go2 => true.
  
  
model2() =>
  N = 3,

  X = resample(N, 1..6),
  S = X.sum,
  
  add("s",S).


/*
  """
  (c) If we instead draw three cards without replacement (so the probabilities are no longer 
      independent), what is the average value of the total?
   """

  This is also the same estimated value. Though this has has a little different
  probability distribution than for a) and b).

  var : s
  Probabilities:
  10: 0.1527000000000000
  11: 0.1523000000000000
  12: 0.1508000000000000
  9: 0.1467000000000000
  13: 0.1029000000000000
  8: 0.0989000000000000
  14: 0.0498000000000000
  6: 0.0496000000000000
  15: 0.0482000000000000
  7: 0.0481000000000000
  mean = 10.5156


*/
go3 ?=>
  reset_store,
  run_model(10_000,$model3,[show_probs,mean % ,
                           % show_percentiles,
                           % show_hpd_intervals,hpd_intervals=[0.94],
                           % show_histogram,
                           % min_accepted_samples=1000,show_accepted_samples=true
                          ]),
  nl,
  % show_store_lengths,nl,
  % fail,
  nl.
go3 => true.
  
  
model3() =>
  N = 3,

  X = draw_without_replacement(N, 1..6),
  S = X.sum,
  
  add("s",S).