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[Fingers]

You're on a game show. The host offers you three boxes. In one is a set of keys to a new car. The other two are empty.

You choose a box.

The host (who knows which one has the keys in) opens one of the other boxes and shows you it is empty. He now offers you the chance to change your mind and switch to the other box, or stick with your original selection.

Are you better off changing your mind, or sticking with your first selection?


PS. You may have seen this before in which case forgive me for boring you. I just find the answer tends to surprise people.
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[RiotNrrd]

I *have* seen this before, but I've forgotten the answer, so it's almost like the very first time.

[ronbo27]

Switch???
Cheers!
Ron.
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[RiotNrrd]

To reassure myself that my answer was correct, I wrote down all the possible outcomes. Even then, it took a bit to realize why the correct answer was actually correct, as there is an aspect to it can be misleading (which is, no doubt, why the puzzle is a puzzle and not one of those 1+1=2, duh, sorts of things).

[bobkart]

Hopefully I am not spoiling Fingers' puzzle by posting this hint.

In this puzzle, N=3 (there are three boxes). It's easier to see why the answer is what it is when you increase N, say to 1,000,000. The keys are in one of a milion boxes, and you pick one, then the game show host opens all but one of the remaining boxes to show you that they are empty. Now what seems like the right thing to do?

[RVNOak]

I know nothing about this and will wait here for the answer...
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[Downtown_Charlie]

does anyone hear rember Michael Miles and Take Your Pick?
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[JohnnyAardvark]

[JohnnyAardvark]

[phillipthorne]

Michael Miles - black and white TV - Take your Pick - peak time viewing. Goodness I feel old !!!! Who can forget the Yes No Interlude and Hughie Green on Double Your Money. Now I feel REALLY old !!

Don't tell me you remember it too D.C. ?? You're not going to reminisce about the Flower Pot men and Muffin the Mule as well are you

http://www.whirligig-tv.co.uk/tv/adults/quiz/quiz.htm for a trip down memory lane.

Best wishes
Phillip
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[RiotNrrd]

your odds are still pretty much 50/50

Heh. Actually, they're not.

To solve the problem, pretend the key is in box #3. Write down all your possible actions:

Take box #1 and hold it. Win or lose?
Take box #2 and hold it. Win or lose?
Take box #3 and hold it. Win or lose?

or

Take box #1 and switch. Win or lose?
Take box #2 and switch. Win or lose?
Take box #3 and switch. Win or lose?

Since there are three possible actions, the odds cannot be 50/50. 2/3 of the time one thing will happen, and 1/3 of the time something else will happen. You want the good event to happen 2/3 of the time. If you write down the answers to the above questions you will see which strategy is best.

[Fingers]

As always it's a close run thing on the poll but the actual answer is:

You are better of changing your mind when given the choice. If you change your mind you have a 2 in 3 chance of winning.

The explanation above is absolutely right.

It's really easy to understand if you sketch it out.... so I have....



So in actual fact it's worth thinking off this in the negative....

The only time you would be worse off changing your mind is if you picked the correct box first. The odds of doing that were 1 in 3....

Obvious when you think about it - but always seems surprising to me!
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[JohnnyAardvark]

[RiotNrrd]

You are left with two boxes... One is right and one is wrong. You have chosen one of those boxes... Thats what I was basing the not far off 50/50 on.

That's how most people think about it (including myself when I first heard this puzzle - I also initially swore the odds were 50/50 because the final choice involves only two boxes; that's what I meant about the misleading aspect of the puzzle). It's only when you sketch it out that you see that if you always choose to switch your odds are go from 50% to 66 2/3%.

The difference between 1/2 and 2/3 actually is quite substantial. In comparison, a roulette wheel gives the casino only a 55.25/44.75 advantage (because of the 0 and 00 spaces on the board), and about the only standard casino gambling game that gives you a worse payout than that are the slot machines (which practically rob you, with margins sometimes over 30% in the casinos favor - the term "one armed bandit" isn't inaccurate at all). Yet that 5.25% advantage at roulette makes the casinos many many many millions per year. Craps and blackjack have better odds, but still tip in the casinos favor a bit and that's all it takes to make them rich.

Of course, casinos have the advantage of the "long haul", over which random fluctuations (i.e., "luck") even out mathematically. In this puzzle you get just one shot, so you can still lose even when you adopt the winning strategy. Nevertheless, I'd take any advantage I could get!

[bobkart]

Simply put, it's only wrong to switch if your first choice was correct, the odds of which are 1 in 3.