The Monty Hall problem (as famously debated here and elsewhere) is the counter-intuitive situation when having made a random choice, additional information is given that seems to alter the odds that one guessed correctly when in fact it does not. I was wondering whether the game show Deal or No Deal is a Monty Hall situation or not. Like Monty Hall, a contestant starts by making a random guess (here one out of 26), and then information is given about which choices one did not make, and re-estimating one’s odds based on that info.
Now on DoND one’s last chance to cash out comes before the last case is opened, but let’s assume that wasn’t the case and instead you got down to one unopened case: you now know that you picked either the million-dollar case or the 1-cent case. Are the odds fifty-fifty that you guessed right- or one in 26? Compare this to a Monty Hall situation where there are 26 cases, one with a million dollars and 25 with one cent. After you guess, 24 cases are opened revealing one cent. How is this different from the idealized Deal or No Deal scenario?
It seems to me that when the show starts, you have a 1/26 chance of having guessed correctly, and that as the other cases are revealed it goes down to 1/25, 1/24, etc (unless of course the million dollar case is revealed and your chances go to zero). So at the end, you have a 1/2 chance of being right and to switch or not is no advantage either way.
Monty knows which door has the good prize and intentionally reveals one that doesn’t. So originally you have a 1/3 chance of being right, trading doors increases that to 2/3.
It’s 50/50 for the Deal or no Deal case. The key difference is that you are the one selecting the cases to open, and have no prior knowledge. There are many, many scenarios in which one of the cases you open will have the million. The fact that you happen to be in one of the scenarios where you didn’t doesn’t change the final odds.
In the Mony Hall problem Monty opens the doors, and he never shows you the million, always the cents. So now there are no possibilities where you would have accidentally opened the wrong case, as Monty would never do that. So all of those odds “collapse” in to the other unopened case (which you should definitely switch to).
I agree with you both. The dice have no memory. I’ve you’ve picked 25 cases already and haven’t hit the million, you’ve been lucky. But that doesn’t change the odds from 50/50.
The key to the Monty Hall problem is in the assumptions: It has to be done by someone who knows which door to open. It also has to be done consistently. If they only did it in cases where it would be to your detriment, it could easily be good to turn it down. It’s only when it’s consistently done that the odds will help you.
James May did a nice demonstration of this using a russian roulette scenario (where one of three cans of beer was shaken up). In 100 trials, the benefits of switching played out exactly like the statistical predictions suggested!
Er, isn’t it more like the other way around? The additional information seems not to alter the odds, but in fact it does.
In Deal or no Deal I suppose you might be getting odds-affecting information if the Banker (?, the hidden guy who offers you the deals) has some foreknowledge of what is in the cases. That might affect his offers, and the contestant might be able to infer something from that. However, I would assume that the Banker is as much in the dark as the contestant is.
There may be a bit of a paradox because of that: the more your opponent (Banker) knows, the better your chances might be of a big win, because he is likely to ‘leak’ useful information to you in the very act of trying to thwart you. (I am not sure if it would work like that, but I think it might.)
The additional information doesn’t alter the odds that your initial guess was correct: that’s still a 1/3 probability.
Since Monty always opens a door with a goat, no matter what you originally picked, the fact that he does so doesn’t provide you with any additional information about the door you picked; it only tells you something about the doors you didn’t pick. This is not the case with the Deal or No Deal scenario: once you pick your case, having all the other cases except the million get opened is not an inevitability.
In the Deal or No Deal situation, the host doesn’t know which case holds the prize.
In the Let’s Make a Deal situation, if Monty didn’t know which door it was and picked a random door to reveal, just getting lucky that he didn’t reveal the prize, then it would be the same situation as Deal or No Deal - your odds would be 50/50.
The Monty Hall puzzle solution of 2/3 depends on Monty knowing and consistently revealing a non-prize door.
I don’t get the appeal of Deal or No Deal myself - it’s not a game that rewards intelligence, skill, knowledge or even guts, really. It can only maintain some level of tension as long as the contestant avoids randomly opening the big-money amounts.
It has already flamed out, but the drama of seeing whether someone takes a deal or turns it down only to wind up with less was a lot of fun. So that’s one reason. Then there are 26 models so there’s 26 other reasons.