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You're on a game show, playing for 1 million dollars! You've made it to the final round, there are 4 doors, you pick a random door (obviously, there's a 1/4 chance for all of them, let's say you picked #3.) The host reveals door #1, it's empty. You now have the choice to stick with your door, switch to another door, or take 5 hundred thousand dollars! Which should you take?

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    $\begingroup$ Seriously, if I had the choice between just taking half a million or I had a 1 out of 3 chance of getting nothing... it should be clear to anyone except the greedy. $\endgroup$ – generalcrispy Oct 21 '14 at 12:56
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    $\begingroup$ You're not saying what the strategy of the host is, i.e. why he chooses door #1. Therefore the answer is as undefined as with all those mis-told versions of the Monty Hall puzzle. Please elaborate. $\endgroup$ – Kilian Foth Oct 21 '14 at 14:31
  • $\begingroup$ I agree with general crispy: choose a door: significant chance of being sad: take 1/2 million: zero chance of being sad. Easy choice. $\endgroup$ – user3294068 Oct 23 '14 at 15:16
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Note: I assume the host knew door #1 would be empty. This is analogous to the original Monty Hall problem. If the host randomly opens a door, the door-opening is moot and can be left out completely.

Let's first look at the situation without the 500K consolation prize:

Let's say I do NOT switch doors, then I have a 25% chance to win 1 million.

Let's say that I do switch:

  1. I pick the right door immediately (25% chance). Switching wins me nothing.
  2. I pick the wring door first (75% chance). After the elimination of a wrong door, I switch with a 50% chance to win 1 million.

I have in total $\frac{3}{4} \cdot \frac{1}{2}=\frac{3}{8}$ chance of winning the 1 million.

So switching is, like in the original Monty Hall problem, certainly beneficial, raising my chance of winning from $\frac{1}{4}$ to $\frac{3}{8}$.

However, my expected win is $\frac{3}{8} \times \text{1 million}$. That is less then the 500K.

Basically, I have a choice to gamble a sum of money with the chance of doubling it, but I have only $\tfrac{3}{8}$ chance of doubling it. Going by the math, I should stick with the money I have, which means
I should choose to keep the 500k.

This problem introduces something that is not in the original problem, and that is utility. That is the phenomenon that what I can win changes my life much more than the money I loose. It is the reason people play in lotteries with a negative expected return (winning some millions changes your life. Paying some dollars every month makes no change. So even if I never win, the possible win is much more important than the almost certain loss).

Now, in the case of "investing" 500k at bad odds to double it, this doesn't seem to play (both 500k and 1 million are "a lot of money!), but let's say I have a loan shark on my case who will kill me if I do not pay him back 1 million tomorrow. Now, the 500k will do me no good at all. In this situation, I will certainly select another door, giving me a 3/8 chance of surviving tomorrow. (Versus taking the 500k which will see me dead, or not switching and sticking with my original door, which gives me 25% chance of survival).

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  • $\begingroup$ Awesome job! Just the answer I was looking for! $\endgroup$ – warspyking Oct 21 '14 at 13:18
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    $\begingroup$ +many for mentioning the utility of money - which so many of these problems ignore! $\endgroup$ – Julia Hayward Oct 21 '14 at 13:26
  • $\begingroup$ @JuliaHayward honestly, how important is that? I mean, it's an interesting thought, but do we really need to add "but what if there is a crazed gunman demanding one million dollars" to every question? It's safe to assume each question wants a strategy maximizes expected dollar outcome, unless a utility breakdown is explicitly stated. $\endgroup$ – TheRubberDuck Oct 21 '14 at 18:59
  • $\begingroup$ @EnvisionAndDevelop: The guntoting madman is only relevant when there is a small difference in order of size between the prizes. Utility only comes into play when it influences the strategy. It is an essential difference between the original MH problem (which is always, in any strategy, an all-or-nothing game) and this variation (where I may have the opportunity to select my strategy based on utility). A more common occurrence of utility is indeed in standard lotteries, where the expected result is negative, and yet people play (small loss = no change, but big win = big change). $\endgroup$ – oerkelens Oct 21 '14 at 20:10
  • $\begingroup$ I'm moreso commenting on the arbitrary inclusion of an extra factor. You could have just as easily included a change in strategy for when the madman will steal money from you if you swap, or if you choose 3, his unlucky number. Why include one but not the other? I suppose you can include what's interesting to you, but I was responding to the sentiment that it's something people should be rewarded for doing. $\endgroup$ – TheRubberDuck Oct 21 '14 at 20:26
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Take the $500,000.

I assume that opening a door for an empty room means you get nothing.

You picked on, and one was eliminated. So, there is now a $1/3$ probability of picking the correct door from the remaining 3. \$500,000 $> 1/3*1000000$.

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  • $\begingroup$ This is not true. The chance is 1/4 * 1000000. See the Monty Hall problem. The answer is still right, but the calculation isn't $\endgroup$ – Mathias711 Oct 21 '14 at 12:58
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    $\begingroup$ Yes, @Mathias711: see the Monty Hall problem. The chance is 3/8 to win the million, not 1/4 (if you switch, of course) $\endgroup$ – oerkelens Oct 21 '14 at 13:11
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I would take the money , because its higher than the probability of winning the chance which is 1/3rd (Edit: its 3/8 as explained in the other answer), but that is assuming the host knows which door it is in.

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  • $\begingroup$ The chance of winning is 1/4th. But yes, keep the money! $\endgroup$ – Mathias711 Oct 21 '14 at 12:53
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    $\begingroup$ @Mathias711 It's 1/3 because an empty room has been eliminated $\endgroup$ – d'alar'cop Oct 21 '14 at 12:55
  • $\begingroup$ puzzling.stackexchange.com/questions/2167/… $\endgroup$ – Mathias711 Oct 21 '14 at 12:56
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    $\begingroup$ @Mathias711 You are making an assumption that the host knew door #1 was empty. That has not been stated here. $\endgroup$ – Joel Rondeau Oct 21 '14 at 12:58
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    $\begingroup$ The probability of winning with switching is not 1/4. It is 3/8. $\endgroup$ – oerkelens Oct 21 '14 at 13:08

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