Making the Monty Hall problem weirder but obvious(dyno-might.github.io)

  A_x = "contestant first picks door x"
  B_y = "Monty opens door y, revealing a goat"
  C_z = "price is behind door z"

  P(C_z) = 1/3, the price can be behind any door with equal probabilities
  A and C are independent, the contestant has no prior knowledge of the placement of the price
  P(B|A&C_3)=1, that is Monty opens door 2 with probability 1 if the contestant first opened door 1 and the price is behind door 3, Monty deliberately picks the door with the goat, very important!
  P(B|A&C_2)=0, Monty never opens the door with the price.
  P(B|A&C_1)=1/2, Monty equally randomly picks between two doors when he can.

  P(C_3|A & B) //probability for winning when switching
  = P(A & B|C_3)*P(C_3) / P(A & B)
  = P(B|A & C_3)*P(A|C_3)*P(C_3)
    /(P(B|A)*P(A)) // P(A|C_3) = P(A) due to independence
  = P(B|A & C_3)*P(C_3)
    /P(B|A) //expand denominator
  = P(B|A & C_3)*P(C_3)
    /( P(B|A & C_1)*P(C_1)
       + P(B|A & C_2)*P(C_2)
       + P(B|A & C_3)*P(C_3) ) // use P(C_1) = P(C_2) = P(C_3) = 1/3
  = P(B|A & C_3)
    /( P(B|A & C_1)
       + P(B|A & C_2)
       + P(B|A & C_3) ) // substitute all our assumptions above
  = 1 / (1 + 0 + 1/2)
  = 2/3

  In front of you are 3 doors, there is a goat behind two 
  of them, and a car behind the other one. 

  In *round 1* of the game, you select a *pair* of doors,  
  from which *one* “goat containing door” will be
  *automatically eliminated from*, leaving only *one* door
  of the selected pair of doors in play, (and only *two* of
  initial *three* doors in play).

  In round 2 of the game, you guess which of the two
  remaining doors in play has the car.

    import numpy as np
    
    def play_monty_hall(rounds=100):
        car = np.random.randint(low=1,high=4,size=rounds)
        first_door = np.random.randint(low=1,high=4,size=rounds)
        switching_wins = car != first_door
        staying_wins = np.logical_not(switching_wins)
        print(sum(switching_wins))
        print(sum(staying_wins))

   {
      int count = 0;
      int i;

      for (i = 0; i < 1000; i++) {
         int choice1 = rand()%3;
         int actual = rand()%3;
         if (choice1 != actual)
             count++;
   }