Monty Hall Problem: Alternative assumptions

Evil Monty: He only asks you to switch when he knows you have the prize… in that case never switch.
Ignorant Monty: He doesn’t know where the prize is! In that case he might open the door with the prize. Using the same methods we used above it can be shown that the probability you have the prize is now 0.50 and it doesn’t matter if you switch. Let’s sim!

num_samples = 10000
monty_1 <- dplyr::tibble(
   door_with_prize = sample(c(1,2,3), num_samples, replace = TRUE),
   door_monte_opens = sample(c(2,3), num_samples, replace = TRUE) 
)

# We observe open door is #2 and no prize there: 
monty_1 |>
   dplyr::filter(door_monte_opens == 2 & door_with_prize != 2) |>
   dplyr::summarize(mean(door_with_prize == 1))

## # A tibble: 1 × 1
##   `mean(door_with_prize == 1)`
##                          <dbl>
## 1                        0.510

For more on this probem and the media furor, see: https://en.wikipedia.org/wiki/Monty_Hall_problem.