/* 

  Second chance in Picat.

  https://brainstellar.com/puzzles/probability/114
  """
  Roll a die, and you get paid what the dice shows. But if you 
  want, you can request a second chance & roll the die again; get paid what 
  the second roll shows instead of the first. What is the expected value?
  """
  
  Cf my Gamble model gamble_second_chance.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.

main => go.

/*
  If we roll again if first roll is below or equal a limit then the best limit is
  3 with an expected outcome of 17/4 (4.25)

  limit = 1
  var : roll
  Probabilities:
  3: 0.1999000000000000
  6: 0.1972000000000000
  4: 0.1957000000000000
  2: 0.1910000000000000
  5: 0.1859000000000000
  1: 0.0303000000000000
  mean = 3.9075


  limit = 2
  var : roll
  Probabilities:
  5: 0.2235000000000000
  6: 0.2232000000000000
  4: 0.2216000000000000
  3: 0.2182000000000000
  1: 0.0590000000000000
  2: 0.0545000000000000
  mean = 4.1657


  limit = 3
  var : roll
  Probabilities:
  6: 0.2561000000000000
  4: 0.2518000000000000
  5: 0.2400000000000000
  3: 0.0866000000000000
  1: 0.0828000000000000
  2: 0.0827000000000000
  mean = 4.2518


  limit = 4
  var : roll
  Probabilities:
  5: 0.2768000000000000
  6: 0.2752000000000000
  4: 0.1162000000000000
  3: 0.1121000000000000
  1: 0.1107000000000000
  2: 0.1090000000000000
  mean = 4.165


  limit = 5
  var : roll
  Probabilities:
  6: 0.3052000000000000
  1: 0.1430000000000000
  5: 0.1400000000000000
  3: 0.1389000000000000
  2: 0.1370000000000000
  4: 0.1359000000000000
  mean = 3.9085


  limit = 6
  var : roll
  Probabilities:
  3: 0.1708000000000000
  6: 0.1683000000000000
  4: 0.1673000000000000
  1: 0.1649000000000000
  5: 0.1644000000000000
  2: 0.1643000000000000
  mean = 3.5069


*/
go ?=>
  member(Limit,1..6),
  println(limit=Limit),
  reset_store,
  run_model(10_000,$model(Limit),[show_probs_trunc,mean]),
  nl,
  % show_store_lengths,nl,
  fail,
  nl.
go => true.
  
  
model(Limit) =>
  N = 6,

  RollDice = random_integer1_n(N,2),

  Roll1 = RollDice[1],
  Roll = cond(Roll1 <= Limit, RollDice[2], Roll1),
  
  add("roll",Roll).


/*
  Decide to roll again by chance (50-50).

  var : roll
  Probabilities:
  6: 0.1730000000000000
  5: 0.1674000000000000
  1: 0.1669000000000000
  4: 0.1658000000000000
  3: 0.1650000000000000
  2: 0.1619000000000000
  mean = 3.5239

*/
go2 ?=>
  reset_store,
  run_model(10_000,$model2,[show_probs_trunc,mean]),
  nl,
  % show_store_lengths,nl,
  fail,
  nl.
go2 => true.
  
  
model2() =>
  N = 6,

  RollDice = random_integer1_n(N,2),

  Roll1 = RollDice[1],
  Roll = condt(flip(1/2), RollDice[2], Roll1),
  
  add("roll",Roll).