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When I read this question (problem restated below), and the first comment, I was drawn to the great similarities between this problem and the Monty Hall problem (asking for the winning probability if you switch). Both include a choice with partial information about what's hidden behind the doors / in the compartments. In both cases it's very easy for lay-people to assume the probability is $1/2$, when in fact it's $2/3$.

Is there an easy translation between the two problems? What do the goats and car correspond to in the boxes-with-gold-and-silver-bars problem?

I have some ideas, stated in an answer below, but it's far from complete, and I would like to hear other ideas as well. If there is some abstract, literal translation then I would love to hear it.


I have three boxes, each with two compartments.

  • One has two gold bars

  • One has two silver bars

  • One has one gold bar and one silver bar

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  • You choose a box at random, then open a compartment at random.

  • If that bar is gold, what is the probability that the other bar in the box is also gold?

enter image description here

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This isn't a full translation, but it's half-way.

Say, in the metal-bars-problem, that you're after the double-gold box. Restate it like this: you pick one box, and then open a random compartment in one of the other boxes. This turns out to be gold. What is the probability that you get the double gold if you switch to that box?

This immediately looks more like the Monty Hall problem, but instead of opening a full box that isn't double-gold, we're opening half a box and seing one gold bar.

In some sense we're told that the half-opened box is most likely more valuable than the unopened box you didn't pick, which is exactly what mr. Hall tells us when he opens a door with a goat: telling you which of the two other doors is probably more valuable.

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You would have a probability of $\frac{2}{3}$ if you switched boxes the same as the Monty Hall scenario.
The half opened box is decidedly more valuable than the unopened one in the sense that it conveys information which you can use to make your decision, which is analogous to the door that Monty opened.
There are a few interesting variations on this theme.

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Gold Bars: There are three bars of gold could have found by opening a compartment, of which two are in the same box as each other.   If you had selected the compartment without bias, then the conditional probability that you chose the box with two gold bars is $2/3$.

Monty Hall: There are three doors you could have opened, of which two hide a goat.   If you had selected a door without bias, then the probability that you will win a car by switching on condition that Monty reveals a goat is $2/3$.

General: Under a condition that eliminates options, there are three outcomes that may have happened without bias between them, of which two are labelled "success" and the third labelled "fail".   The probability of a success is $2/3$.

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