Deal or no deal: does one switch (to avoid a goat)?/ Should deal or no deal be 10 minutes shorter?

Question

Okay so this question reminded me of one my brother asked me a while back about the hit day-time novelty-worn-off-now snoozathon Deal or no deal.

For the uninitiated:

In playing deal or no deal, the player is presented with one of 22 boxes (randomly selected) each containing different sums of money, he then asks in turn for each of the 21 remaining boxes to be opened, occasionally receiving an offer (from a wholly unconvincing 'banker' figure) for the mystery amount in his box.

If he rejects all of the offers along the way, the player is allowed to work his way through several (for some unfathomable reason, emotionally charged) box openings until there remain only two unopened boxes: one of which is his own, the other not. He is then given a choice to stick or switch (take the contents of his own box or the other), something he then agonises pointlessly over for the next 10 minutes.

Monty hall

[If you have not seen the monty hall 'paradox' check out this wikipedia link and prepare to be baffled, then enlightened, then disappointed that the whole thing is so trivial. After which feel free to read on.]

There is a certain similarity, you will agree, between the situation a deal or no deal player finds himself in having rejected all offers and the dilemma of Monty's contestant in the classic problem: several 'bad choices' have been eliminated and he is left with a choice between a better and worse choice with no way of knowing between them.

So???

Question: The solution to the monty hall problem is that it is, in fact, better to switch- does the same apply here? Does this depend upon the money in the boxes? Should every player opt for 'switch', cutting the 10 minutes of agonising away???

Answer 1

I would ceratinly plump for saying they were opened at random

Then no, the Monty Hall solution doesn't apply. The whole point is that the door isn't randomly opened, it's always a goat.

An easy way of seeing this is imagining there are 100 doors, with 99 goats. If, after you pick a door, the host always opens 98 doors of goats, then switching is very intuitively favorable. However, if he had just opened 98 doors at random, then most of the time (98 out of 100) he would open the door with the car behind it; and even on the rare occasions he didn't, you still wouldn't be any better off switching than staying.

See also this answer, in which I try to intuitively explain probability fallacies.

Answer 2

This doesn't have the key features of the Monty Hall problem:

1. The contestant makes a choice
2. The contestant is given information about other choices they could have made
3. The contestant can change their choice based on this new information

Without being given extra information, their is no point in changing your choice.

Answer 3

22 boxes. 2 at the final stage.

-Assume you will definately swap. If you've picked 250k, which is 1/22 chance, you lose. If you've not picked 250k, which is 21/22 chance, you win. Every 22 times you play, you would expect to pick 250k once and expect to have 250k in the last 2 boxes twice (22 x 2 = 44). Of the two times that 250k is in the last 2 boxes, 1 time you will swap (and lose), 1 time you will swap (and win).

-Assume you will definately not swap. If you've picked 250k, which is 1/22 chance, you win. If you've not picked 250k, which is 21/22 chance, you lose. Every 22 times you play, you would expect to pick 250k once and expect to have 250k in the last 2 boxes twice (22 x 2 = 44). Of the two times that 250k is in the last 2 boxes, 1 time you will not swap (and win), 1 time you will not swap (and lose).

If you swap (or not), half the time you will win, half the time you will lose. With the monty hall problem, 2/3 or the time if you swap you will win, 1/3 of the time you will lose. The difference is that in EVERY game of the MH problem, a losing box is revealed and the player reaches the final stage, even though there are more than 2 boxes to start with. In deal or no deal, 250k would only be at the final stage 2/22 or 1/13 times.

Answer 4

The key difference between the Monty Hall problem and the one you're describing is: a) the host (in the case of Monty Hall) makes an unsuccessfull choice for the contestant thus altering the probabilities. b) the host can't provide that information when there is only a binomial distribution (2 choices remaining). If he did your probability would go from .5 to 1. With Monty Hall your probability goes from 1/3 to 2/3 because you have 2 in 3 chances of NOT choosing the goat.

Answer 5

I don't think it matters how you get to the final 2 figures, the question is out of a pool of 20 boxes, or however many it is, what are the chances that I picked the 250k box. The odds are 1 in 20, and that doesn't change. If it's 1 in 20 that I hold the main prize, and it's certainly in 1 of the boxes, then the other box has to have a 19/20 chance of being the top prize (as the chance of it being in one of them is 1/1).

Think of it like this - if it were a 50/50 chance at 2 boxes, you would need to be able to guarantee selection of that box 50% of the time out of 20 boxes - impossible. In 19 out of 20 scenarios, either me picking 1, 2, 3 etc, I only won the 250k prize if I swap, random or with host knowledge is irrelevant.

Something else occurred to me as well - The object of the game is not necessarily to win the 250k, but the highest amount possible. If after the game is played out, I am left with 1p and £2000, I can assume that the original pool I was selecting contains everything between those 2 figures and none of the greater amounts, and adjust the odds accordingly. If there is a 1p, a 50p, £1, £5, £10, £50, £100, £500, £1000 and £2000, and I know with 2 boxes at the end of the game that a 1p and a £2000 are left, then it must still hold that out of those 10 boxes, 9 times I will pick something other than the £2000 box. The fact that I picked the 1p isn't important – it could have been any amount that isn't equal to and below the £2000. In 9 out of 10 cases the £2k is in the other box and I only win the LARGEST AMOUNT I CAN if I swap (given the choice).