Deal or No Deal TV show reporting odds …. correctly

UPDATE: Ann's clever simulations indicate the odds are as stated on the show.

The TV Show "Deal or No Deal" appears to be a complex variation on the Monty Hall problem, but it's NOT.  I'm deleting my earlier *failed logic* which was wrong.

Apparently Monty's intentions matter in the 3 door game as follows:

Contestant picks door – this door will NOT be opened.

THUS there are 3 possible ways for the remaining 2 doors to be opened:

Empty, Empty
Empty, Prize
Prize, Empty

In Monty Hall, he KNOWS where the prize is and always uncovers an empty door.    However in a game where the host does NOT know and thus the prize CAN be uncovered, the game universe will EXCLUDE the Prize, Empty option.   This exclusion changes the odds, with each remaining option having a 1/3 chance of happening = equal odds whether you switch or not.

15 thoughts on “Deal or No Deal TV show reporting odds …. correctly”

1. I think Wikipedia is wrong, too, exactly as you say. I just got into a heated debate about this at a Channel 4 discussion forum. But the Wikipedia source says they tested it. They only thing I can think to do now is to run a simulation to prove it to myself and work up enough courage to correct the Wikipedia entry.

2. It *has* to be 50:50 because the final 2 cases are completely random. There is nothing at all to suggest that the big money is more likely to be in one rather than the other.

3. Ann’s simulation sure looks good and does support 50/50. But Phil why is this NOT a Monty Hall variation, where you should switch your pick after an empty case is revealed to increase your odds of winning from 1/3 to 2/3?

4. HOWEVER – The case that is sitting on your desk (unless the 1 million is removed from the board) has a consistent 1/26 chance of holding the 1 million. Those aren’t very good odds. So when you get to the end, it still has a 1/26, right? If you have only the one on your desk and 1 million on the board, then you would want to go with the other one, (theoretically 25/26) wouldn’t you?

5. Paul that was my first thought dut to Monty Hall but no, it is a 50/50 chance as they say on the show. Unlike Monty, the odds you have the million in Deal or no deal change as you remove cases from the game. I’m not clear why Monty and this are different though.

6. Its different because montys choice was not a random selection. Imagine choosing from 1,000,000 cases, the odds that you choose the million \$ case is 1/1,000,000. After you miraculously eliminate the other 999,998 cases, do you really think that the odds that the last case is the million is 999,999/1,000,000? The difference is that it would be a miracle if you randomly selected all the cases and didnt pick the million. In the other case, it would be no miracle, it would be a gaurentee that monty opens up all the wrong cases.

The odds change as more cases are revealed. A good example for any poker player would be this. Say you start with pocket aces and you go all in against pocket 5’s. Your odds of winning are roughly 80% right? But once more information is revealed your odds of winning change if he got a flush draw.

Your chance of having the case at the begining is one out of 26, when there are 25 cases, its 1/25 and so forth.

7. your chances of picking the million is very weak, its all compound probability. When u pick your case your chance of a million is 1/26. then when u pick the st case to eliminate and it is not the million your chances at the million are 1/26 * 1/25 and on the next case its 1/26 * 1/25 * 1/24 and so on until you get to the final 2. If you do get to the final two cases it is very wise to switch as the last case you have no eliminated has a 1/2 chance of being the million and the case you picked has a much much much smaller chance of being the million….

8. Jeez! Why don’t ppl get this?

It’s 50:50 due to compound interest attained from you taking out the other boxes.

In the Monty case he will ALWAYS choose an empty door to open, so your chance of wining with your chosen door is 1/3 but the chance of winning by switching (after the other door has been opened) is 2/3.

The main differences is that you will always get to this stage with Monty. With DOND there’s a big chance you will eliminate the top prize before you get to the stage that requires you to swap. If you do manage to get to the stage that has your chosen box and one other with ANY AMOUNT then the chance of that amount in either box is 50:50.

9. I don’t think it matters if the Host knows where the empty boxes are only that he does open an empty box.

Once that has happened the contestants chances of winning are improved from 1/3 to 3/2 by swapping.

Of course there is a 1/3 chance that an uninformed Host will open the box with the prize but if this happens we never get to the option to swap and so it is not relevant to the swap no swap decision.

Guy

PS where can I find out about “Ann’s clever simulations”?

10. sorry, that should say:-

Once that has happened the contestants chances of winning are improved from 1/3 to 2/3 by swapping.

11. Not sure of the original debate but be to straighten things out. Deal or no deal is not Monty Hall myth.. Monty Hall paradox works if the host KNOWS the location of the prize and the contestant GETS to switch cases. Since neither of these is true, Monty Hall conception doesn’t apply.

The Monty Hall paradox works mainly off your chances to be wrong. In the paradox you pick a door. Monty is then forced to reveal a door different to yours that is not the prize door. Look at the odds you are given after your initial pick. If you picked the prize door the act of “switching” guarantees you lose. If you picked a non-prize door the act of “switching” guarantees you win. Essentially the act of “switching” unlocks a binary absolute outcome. Now consider your odds at the beginning, given all of the above you would always switch because your likely to pick the initial case wrong 2/3 of the time. By switching you flip your odds from 1/3 chances to win to essentially 2/3.

Deal or no deal works mainly off of probability theory. Ideally, each round you have an “expected value” of your remaining cases. This is just a fancy way of saying your average suitcase value. If they offer a value under this expected value then you can expect statistically to come out better by removing a case. However, if they offer you more than your expected value, you are risking your guaranteed amount on chance. Essentially, a truly rational person would take what they’ve won and be content.

Odds of winning the million on the other hand are different. You’re odds can be no worse than 1/26. Whoever said above that your odds gets multiplied as cases are revealed is wrong as this would give you a great deal less chance each time you pick a new case, which is counterintuitive. .Conditional and compound probability mathematics do not apply. Conditional probabilities work off of the fact that there might be prior knowledge which would affect the probability. Assuming you have american coins, if I told you I had a brownish one in my hand you would know that I’m more likely holding a penny as compared to if I gave you no knowledge. Compound/joint probabilities are probabilities that attempt to quantify the likelihood of two events occurring.

The odds of winning the million round to round have no bearing on your odds besides revealing that you either lost or there is one less case. After each new case is revealed your new odds of winning seem to improve as it is (1/# of cases). However, since you are not allowed to change your case, revealing cases has no effect on your probability. These are what I would deem “hindsight odds”. Regardless of whether you are on the last case or the first case your original choice was made with 26 total cases so the contestants as a whole can do no better than 1/26

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