/* 

  Binomial basketball problem in Picat.

  From https://reference.wolfram.com/language/ref/BinomialDistribution.html
  """
  A basketball player has a free-throw percentage of 0.75. 

  Find the probability that the player hits 2 out of 3 free throws in a game.
  [free_throwIs2]
    Answer: 0.421875

  Find the probability that the player hits the last 2 of 5 free throws. 
  [q1]
     Answer: 0.00878906
   

  Find the expected number of hits in a game with n free throws. 
  [q2]
    Answer: 0.75 n
  """


  Using PDFs:
  1) Find the probability that the player hits 2 out of 3 free throws in a game.
     Picat> X=binomial_dist_pdf(3,0.75,2)
     X = 0.421875

  2) Find the probability that the player hits the last 2 of 5 free throws.
     Picat> X=binomial_dist_pdf(3,0.75,0)*binomial_dist_pdf(2,0.75,2)
     X = 0.0087890625

  3) Find the expected number of hits in a game with n free throws.
     Picat> X=binomial_dist_mean(10,0.75)                            
     X = 7.5

     or perhaps clearer:
     Picat>  X=binomial_dist_mean(1,0.75)   
     X = 0.75


  The three problems are solved in the models below, using either
  binomial_dist and flips.

  This is a port of my Gamble model gamble_binomial_basketball.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.

main => go.

/*

  1) Find the probability that the player hits 2 out of 3 free throws in a game.
  var : prob
  Probabilities:
  0: 0.58109999999999995 (5811 / 10000)
  1: 0.41889999999999999 (4189 / 10000)
  mean = 0.4189

  theoretical = 0.421875

  Using flips
  var : prob
  Probabilities:
  0: 0.57730000000000004 (5773 / 10000)
  1: 0.42270000000000002 (4227 / 10000)
  mean = 0.4227


  2) Find the probability that the player hits the last 2 of 5 free throws.
  var : prob
  Probabilities:
  0: 0.99150000000000005 (1983 / 2000)
  1: 0.00850000000000000 (17 / 2000)
  mean = 0.0085

  theoretical = 0.00878906

  Using flips:
  var : prob
  Probabilities:
  0: 0.99045000000000005 (19809 / 20000)
  1: 0.00955000000000000 (191 / 20000)
  mean = 0.00955


  3) Find the expected number of hits in a game with n free throws.
  var : expected
  Probabilities:
  8: 0.28620000000000001 (1431 / 5000)
  7: 0.24485000000000001 (4897 / 20000)
  9: 0.18915000000000001 (3783 / 20000)
  6: 0.14674999999999999 (587 / 4000)
  5: 0.05680000000000000 (71 / 1250)
  10: 0.05605000000000000 (1121 / 20000)
  4: 0.01625000000000000 (13 / 800)
  3: 0.00355000000000000 (71 / 20000)
  2: 0.00035000000000000 (7 / 20000)
  1: 0.00005000000000000 (1 / 20000)
  mean = 7.5073

  theoretical = 7.5

  Using flips:
  var : expected
  Probabilities:
  8: 0.28039999999999998 (701 / 2500)
  7: 0.25214999999999999 (5043 / 20000)
  9: 0.18484999999999999 (3697 / 20000)
  6: 0.14949999999999999 (299 / 2000)
  5: 0.06060000000000000 (303 / 5000)
  10: 0.05395000000000000 (1079 / 20000)
  4: 0.01535000000000000 (307 / 20000)
  3: 0.00270000000000000 (27 / 10000)
  2: 0.00050000000000000 (1 / 2000)
  mean = 7.4819

*/
go ?=>
  reset_store,
  println("1) Find the probability that the player hits 2 out of 3 free throws in a game."),
  run_model(10_000,$model1,[show_probs_rat,mean]),
  println(theoretical=binomial_dist_pdf(3,0.75,2)),
  reset_store(), % Don't forget to reset the store when using more than one model.
  println("\nUsing flips"),
  run_model(10_000,$model1b,[show_probs_rat,mean]),

  reset_store(),
  println("2) Find the probability that the player hits the last 2 of 5 free throws."),
  run_model(10_000,$model2,[show_probs_rat,mean]),
  println(theoretical=binomial_dist_pdf(3,0.75,0)*binomial_dist_pdf(2,0.75,2)),
  
  reset_store(),
  println("\nUsing flips:"),
  run_model(20_000,$model2b,[show_probs_rat,mean]),

  reset_store(),
  println("3) Find the expected number of hits in a game with n free throws."),
  run_model(20_000,$model3,[show_probs_rat,mean]),
  println(theoretical=binomial_dist_mean(10,0.75)),
  reset_store(),
  println("\nUsing flips:"),
  run_model(20_000,$model3b,[show_probs_rat,mean]),
  nl.
go => true.

% Find the probability that the player hits 2 out of 3 free throws in a game
model1() =>
  P = 0.75,
  FreeThrow = binomial_dist(3,P),
  add("prob",cond(FreeThrow==2,1,0)).

% Using flips instead
model1b() =>
  P = 0.75,
  FreeThrow = bern_n(P,3).sum,
  add("prob",cond(FreeThrow==2,1,0)).


% Find the probability that the player hits the last 2 of 5 free throws
model2() =>
  P = 0.75,
  FreeThrow3 = binomial_dist(3,P),
  FreeThrow2 = binomial_dist(2,P),
  Prob = cond((FreeThrow3 == 0,FreeThrow2 == 2),1,0),
  add("prob",Prob).


model2b() =>
  P = 0.75,
  FreeThrow3 = bern_n(P,3).sum,
  FreeThrow2 = bern_n(P,2).sum,
  Prob = cond((FreeThrow3 == 0,FreeThrow2 == 2),1,0),
  add("prob",Prob).


% Find the expected number of hits in a game with n free throws. 
model3() =>
  P = 0.75,
  Expected = binomial_dist(10,P),
  add("expected",Expected).

model3b() =>
  P = 0.75,
  Expected = bern_n(P,10).sum,
  add("expected",Expected).