Fellow nerds, are you bored each Thanksgiving listening to relatives discussing the newest reality TV shows? If so, here’s a puzzler for you to do while everyone else is watching football. (Hopefully, you have a mathematically-oriented sibling like I do to discuss this with):

Assume a standard Monty Hall setup.

Here’s the twist: The host wants to make the show exciting, so the host will only sometimes reveal a goat the contestant hasn’t picked (the other times the host will reveal what is behind the contestant’s original door, thus ending the game [for better or worse for contestant]). As in the original problem, the host knows what is behind the doors.

Question 1: What strategy must the host use so that if the host decides to open a non-contestant goat door (i.e., if the game proceeds to the second stage) then the contestant is indifferent between (a) staying with their first door and (b) switching to the only remaining door.

Question 2: Given the strategy in Question 1, how often does the contestant win the car?

Question 3: So, just to finish up, if Monty Hall cares about (a) making the contestant indifferent in the second stage, and only after that (b) not giving away cars, and only after that (c) going to the second stage as often as possible, what is Monty’s optimal strategy?

Enjoy and Happy Thanksgiving!