R Stats: Addition Rule and Monty Hall Problem

Addition Rule

• The addition rule states that the probability of event A or event B happening is the probability of event A plus the probability of event B minus the probability of both events A and B happening together.
  Pr(A or B)=Pr(A)+Pr(B)−Pr(A and B)
• Note that (A or B) is equivalent to (A|B)

Example from Blackjack

• We apply the addition rule where A = drawing an ace then a facecard and B = drawing a facecard then an ace. Note that in this case, both events A and B cannot happen at the same time, so Pr(A and B) = 0.
  Pr(ace then facecard) = 4/52 x 16/51
  Pr(facecard then ace) = 16/52 x 4/51
  Pr(ace then facecard | facecard then ace) = 4/52 x 16/51 + 16/52 x 4/51

Monty Hall Problem

• Monte Carlo simulations can be used to simulate random outcomes, which makes them useful when exploring ambiguous or less intuitive problems like the Monty Hall problem.
• In the Monty Hall problem, contestants choose one of three doors that may contain a prize. Then, one of the doors that was not chosen by the contestant and does not contain a prize is revealed. The contestant can then choose whether to stick with the original choice or switch to the remaining unopened door.
• Although it may seem intuitively like the contestant has a 1 in 2 chance of winning regardless of whether they stick or switch, Monte Carlo simulations demonstrate that the actual probability of winning is 1 in 3 with the stick strategy and 2 in 3 with the switch strategy.
• For more in-depth details click here

Code: Monte Carlo Simulation of Monty Hall Stick Approach

B <- 10000
stick <- replicate(B, {
	doors <- as.character(1:3)
	prize <- sample(c("car","goat","goat"))    # puts prizes in random order
	prize_door <- doors[prize == "car"]    # note which door has prize
	my_pick  <- sample(doors, 1)    # note which door is chosen
	show <- sample(doors[!doors %in% c(my_pick, prize_door)],1)    # open door with no prize that isn't chosen
	stick <- my_pick    # stick with original door
	stick == prize_door    # test whether the original door has the prize
})
mean(stick)    # probability of choosing prize door when sticking

Code: Monte Carlo Simulation of Monty Hall Switch Approach

switch <- replicate(B, {
	doors <- as.character(1:3)
	prize <- sample(c("car","goat","goat"))    # puts prizes in random order
	prize_door <- doors[prize == "car"]    # note which door has prize
	my_pick  <- sample(doors, 1)    # note which door is chosen first
	show <- sample(doors[!doors %in% c(my_pick, prize_door)], 1)    # open door with no prize that isn't chosen
	switch <- doors[!doors%in%c(my_pick, show)]    # switch to the door that wasn't chosen first or opened
	switch == prize_door    # test whether the switched door has the prize
})
mean(switch)    # probability of choosing prize door when switching