My brother and our mate spent about four hours on Monday night running through the Monty Hall Problem trying to wrap our heads around the numbers. My brother was convinced that Deal or No Deal was the same problem as Monty Hall. Our mate was convinced otherwise; I tended to agree with him based on my initial thoughts, but having some experience with the weird unintuitive nature of the Monty Hall Problem I deferred a decision until I’d thought about it some more.
This culminated in the creation of some PHP scripts to run a few zillion trials of both problems. The one for Deal or No Deal was quite simple to conceive and it seemed pretty clear almost straight away that it was not the same thing, especially after I started working on the Monty Hall version. Fortunately I found someone else had already done it in PHP so I just played with that.
Anyway, it’s probably obvious to anyone that knows a lot about conditional probability, but they’re not the same problem.
If anyone cares, the PHP script for the Deal or No Deal stuff is up here.
Without thinking hard at all, in Deal or No Deal the host does not eliminate options in the show.
Or more precisely, the host/show does not undertake behaviour consistent with having more knowledge than the contestant.
Yeh – this was the (obvious) difference that Rob and I both came to pretty much straight away. Will, however, was convinced that it didn’t make any difference to the overall odds – his argument was that it was just a general statistical problem (basically he was taking the “weirdness” of the Monty Hall Problem and applying it to any problem that was similar).
So basically the exercise was trying to prove it to him using numbers, which just took a while to come up with enough ways that he was satisfied (for some reason he didn’t trust my awesome skillz at PHP to come up with a reliable solution despite it’s obvious flawlessness!)
Deal or no deal started out with fairly consistent bank offers, but by the end of season 1 the bank had developed a ‘personality’ of offering significantly lower or higher proportion than statistically average. I was thinking that they were attempting to give the illusion of knowledge and bridge the gap to the monty haul type scenario. That interplay between one’s own ‘luck’ and the manipulation by the bank’s offer was the most interesting part of the show.
Unfortunately the variations died down and the show became less interesting. I haven’t watched it for a couple of years though.
Doubling or halving the statistical bank offer doesn’t make it a monty hall scenario, but it does make it a non-rational problem where decisions are made on the perceived information garnered from the bank offering pattern. It turns into a metagame of bluff and counterbluff. Poker anyone?