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# 020

Game Show - September 12, 2009
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Yeah, well I want to see behind door 2. Was it really a million dollars or was it merely the potential of a million dollars?

Monty Hall Problem for the win!

He forgot to ask if door number 1 was chosen randomly!

And he forgot to ask if he chose door number 1 would he get a million dollars instead?

No, KNO3, door number one was a banana. He's just unlucky.

Like the statistician with his feet in ice water and his head in an oven: On average, I feel fine!

I learned about this in maths just the other day :)

OMG THIS WAS IN 21!!!!!!!!!!

Well bmonk, dunno about you, but I would hide the check for million dollars behind the banana in door number one and laugh. I learned statistics from Fat Tony...

The Monty Hall Problem _really_ comes into play at the end of Deal or No Deal, when the player is asked whether they want to switch suitcases with the single case still out. Here the switch is almost a certainty.

@lurker111: The Monty Hall problem does not apply to Deal or No Deal. Since the cases are completely random, there is no statistical advantage to choosing one over the other.

Not as long as they choose which cases to open. As long as the cases chosen are below average, so the remaining average is higher, it's better to switch, on average. For example, if you know the two cases still left include the \$1,000,000 case, since they don't want to reveal it early, you started with a 1/30 chance of picking it. Now, since it is the only one left, you have a 29/30 chance it's the other one.

The problem comes when the big prize is revealed earlier than that--as it often will be when the cases revealed are randomly chosen. In that case, there is no advantage to switching at any point.

I don't know about Deal or No Deal -- but if a case has just been revealed that is _not_ the one you want, and it was _impossible_ for the revealed case to have been among some 1 or more number of set cases (such as the one you have picked), it is always advantageous to switch to a different case.
The scenario is the same as if someone had specifically chosen to reveal only the improper case. Because they didn't reveal the proper case, you can remove the items in the set of possible outcomes that include them revealing the proper case.
It should be a little more obvious that, even if the cases are randomly chosen, if 27/30 cases are revealed to be the wrong ones, and you know it was impossible for them to reveal #28 (which was chosen completely randomly before the revealing) each time they chose a case, that the correct case is probably #29 or #30. #28 was arbitrarily picked before all the random choosing -- #s 29 and 30 were eliminated down.

At least according to http://en.wikipedia.org/wiki/Monty_Hall_problem#Variants_.E2.80.93_slightly_modified_problems ("Ignorant Monty" version) mouster is right.

It makes sense to me (not that making sense to people has much bearing on the correct solution to this problem.) In that case, the only thing you learn by seeing that the revealed cases don't contain the big prize is that either your case or the other remaining case contain it. Both cases are unopened by pure chance (one because you randomly chose it at the beginning, the other because you randomly chose to leave it till last) and have the same chance of containing the big prize.

Whereas if only cases that don't contain the big prize can be opened, then different cases had different chances of being opened (the winning one has 0 chance, the others have a 28/29 chance) so you have extra information about the remaining case; it might still be unopened by chance, but it's probably unopened because it could not be opened earlier.

(and when I say 'the others have a 28/29 chance', by 'the others', I mean, 'all of the cases that you didn't pick, if you picked the winning one.' Of course if you didn't pick the winning one, the other cases had a 100% chance of being opened. Incidentally, this version of the game would be very boring.)

(It would be less boring if all 29 booby prizes were all goats, wolves and cabbages, and you could win them in addition to whatever was in the final case if you managed to get them all safely across a river in a boat which can only hold two at a time.)

A banana sucks, but I wouldn't mind winning a goat.

I spent so much time researching a month or so before I read this. I saw this ending coming the second I read the first sentence.

I laughed so hard, best one yet. Those damn statisticians and there paradoxes. It's a shame it didn't continue into infinity then he'd be set... Sorta

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