In Deal or No Deal, when you get to the end (just two unopened cases left), you're given the option to switch. If the $1M suitcase is still in play, does the monty hall problem apply... at 96% probability?

We have absolutely gained more information that makes one suitcase preferable over another. It is irrelevant whether the host or random chance opened the other cases, because the information we have gained is the same. If the host were to eliminate all but one other case, it would not indicate that the remaining case held the million, just like the host elimination in Monty Hall does not indicate that which of the remaining doors is the winning pick. The OP scenario is not random. Whether it is the host or the guiding hand of fate is irreverent, the information we have is the same. In both cases, we started with a 1/26 chance of guessing the correct case. And I can prove it. All you have to do is run the scenario 26 times with the same set of cases:


The Million Dollars is on Case 26

Scenario 1:

1.) I choose Case 1.

2.) I open boxes 2-25. The Million hasn't shown up yet.

3.) I swap

RESULT: I win.

Scenario 2:

1.) I choose Case 2.

2.) I open boxes 1, and 3-25. The Million hasn't shown up yet.

3.) I swap.

RESULT: I win.

Scenario 3:

1.) I choose Case 3.

2.) I open boxes 1, 2, and 4-25. The Million hasn't shown up yet.

3.) I swap

RESULT: I win.

Repeat for picking cases 4-25, all of which end in a win result

Scenario 26:

1.) I choose Case 26.

2.) I open boxes 1-25. The Million hasn't shown up yet.

3.) I swap

RESULT: I lose.


If you switch step 3 in these scenarios, obviously you lose 25/26 times instead of winning 25/26 times.

At the end of the game, no matter how you got there, you must be living in one of these 26 scenarios, because as the OP said there is just two cases left, and the million hasn't been shown yet. These scenarios show every possible timeline that could have happened with your set of boxes to put you in the position stated by the OP.

In 25 of the 26, you win if you swap.

In 1 of the 26, you win if you keep your case.

So, tell me, should you swap? The answer seems pretty obvious to me.

/r/askscience Thread Parent