/* 

  Correct given Intelligence in Picat.

  Inspired by 
  https://www.reddit.com/r/Probability/comments/1en5u8e/q_how_to_think_about_probability_of_being/
  """
  [Q] How to think about probability of being right/wrong, considering intelligence distribution?

  Hi folks, it's been a while since I exercised my probability muscle (and truth be told, 
  I was never great with probabilities), so I'm turning to you for help.

  I have the following problem statement: Consider a normal distribution of intelligence. 
  What is the probability of lower-than-average IQ people being wrong/right about any 
  given topic? For simplification, don't take into account the complexity of topics 
  they need to answer.

  Given only the hypothesis above, my naive answer would be that we can consider (for 
  the sake of the problem) intelligence unrelated to being right/wrong. Is p=0.5 just as 
  if it's a fair coin toss, then?

  But what if we try to solve it by making the assumption that intelligence does correlate 
  with being right/wrong? What does the math look like then?

  Thanks in advance!
  """

  The idea in this model is that the higher a person's intelligence is, the probabiity  
  of being correct increases. 
  I'm using the CDF for this correlation: a person correct answers 
  a question with probability flip(CDF...)), i.e.the CDF is the probability to flip.
  This might very well be a doubtful approach.

  Let's assume that we are asking just true/false questions.

  Here are some diffent point values for different values of IQ:

  Picat> X=normal_dist_cdf(100,15,80)
  X = 0.091211219725868

  Picat> X=normal_dist_cdf(100,15,90)
  X = 0.252492537546923

  Picat> X=normal_dist_cdf(100,15,100)
  X = 0.5

  Picat> X=normal_dist_cdf(100,15,120)
  X = 0.908788780274132

  Picat> X=normal_dist_cdf(100,15,130)
  X = 0.977249868051821

  Below are simulations for some ranges.

  Cf my Gamble model gamble_correct_given_intelligence.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.

/*
  * A person with IQ < 100

    [mean = 100,sigma = 15,lower_limit = 0,upper_limit = 99]
    var : iq
    mean = 87.4807

    var : prob correct
    mean = 0.237524

    var : correct
    mean = 0.242903

  * A person with IQ > 130

    [mean = 100,sigma = 15,lower_limit = 131,upper_limit = 400]
    var : iq
    mean = 136.275

    var : prob correct
    mean = 0.990128

    var : correct
    mean = 0.991045

  * A person with IQ between 120 and 130

    [mean = 100,sigma = 15,lower_limit = 120,upper_limit = 130]
    var : iq
    mean = 124.12

    var : prob correct
    mean = 0.943051

    var : correct
    mean = 0.942716


*/
go ?=>
  Presentation = ["iq","prob correct","correct"],
  
  member([Mean,Sigma,Low,Up], [[100,15,0,99],    % IQ < 100
                               [100,15,131,400], % IQ > 130
                               [100,15,120,130]  % 120 =< IQ <= 130
                              ]),
  reset_store,
  run_model(20_000,$model(Mean,Sigma,Low,Up),[mean]),
  nl,
  % show_store_lengths,
  fail,
  nl.
go => true.
  
  
model(Mean,Sigma,LowerLimit,UpperLimit) =>
  IQ = normal_dist(Mean,Sigma),

  ProbCorrect = normal_dist_cdf(Mean,Sigma,IQ),
  Correct = bern(ProbCorrect),

  observe(IQ >= LowerLimit, IQ <= UpperLimit),
  if observed_ok then
    add("iq",IQ),
    add("prob correct",ProbCorrect),
    add("correct",Correct),       
  end.