Variation of Monty Hall problem

There are three envelopes. One contains $50. Another contains $100. The third contains $500. You pick one but it is not opened. If the host reveals one of the other two and it contains $50 or $100 (both considered as goats), the answer is to switch as in Monty hall case. However, if he reveals one with $500, you are now left with either picking one with $50 or $100. Do you switch your choice to the other unopened one? (This was an interview question for a fund manager post)

It depends how Monty decides which envelope to open.

coolkatmeow said:

There are three envelopes. One contains $50. Another contains $100. The third contains $500. You pick one but it is not opened. If the host reveals one of the other two and it contains $50 or $100 (both considered as goats), the answer is to switch as in Monty hall case. However, if he reveals one with $500, you are now left with either picking one with $50 or $100. Do you switch your choice to the other unopened one? (This was an interview question for a fund manager post)

If Monty can reveal the top prize, then we may assume he is opening an envelope at random and this is the so-called Monty Fall problem. In all cases, it makes no difference whether you switch or not.

The Monty Hall problem requires that Monty knows where the car is and never reveals it. Or, in this case, never reveals the $500.

coolkatmeow said:

if he reveals one with $500, you are now left with either picking one with $50 or $100. Do you switch your choice to the other unopened one?

I am not sure I would even call this a variant of the Monty Hall problem. In that one he never showed the top prize, and thus he provided information to the player. Here, since Monty sometimes shows the top prize then that really removes the crux of the scenario.

PeroK said:

If Monty can reveal the top prize, then we may assume he is opening an envelope at random and this is the so-called Monty Fall problem. In all cases, it makes no difference whether you switch or not.

The Monty Hall problem requires that Monty knows where the car is and never reveals it. Or, in this case, never reveals the $500.

Seems like the Monty fall problem is similar to reducing the Monty hall paradox to a two envelopes puzzle (watch YouTube)...yes, in this case the host is just like a second player who does not know where the top prize is, but this second player only have 2 choices to make whereas the first player has 3 options. It's like the first manager has to decide whether the financial market will go up, down or remain flat whereas the second manager just bet that first manager is wrong and choose between the remaining 2 outcomes.

PeroK said:

It depends how Monty decides which envelope to open.

You are exactly right. This is the tricky part because the player or audience does not know whether the host knows where is what and thus observing him revealing any envelope does not affect the probability of the outcome.
If the host already know where is what, then he knows the player made the right first choice or not, but he may intentionally open a goat from the other two doors to mislead the player who thinks the host is giving information to switch choice.

Thanks for all the replies.

I think the best answer for the interview would be to get, say ten players and have them each make their own, independent choices. Then take the track record of the winner and market it, with a 2&20 fee schedule, as an uncorrelated source of alpha