Probability and Death Sentences

[Stephen Tashi]

Labeling the box containing value X, Box 1, as you did for part of the analyses you presented: "Pr(Box 1 is selected) = 1/3", though clearly correct, doesn't require subdivision into the q[1] and [q2] individual probabilities for the revealed coin; doing that neither adds nor detracts from the fact that before and after a coin is revealed, the probability that whichever kind of coin it is, it came from Box 1, is 1/3.

When discussing a problem (given in words) where each outcome may have certain properties, I think the sample space should be detailed enough to exhibit whether each outcome does or does not have each property. If this is not done then the issues of mathematics and the issues of literary interpretation get jumbled together. People make verbal assertions about phenomena. These assertions refer to properties of outcomes. Unless the sample space is detailed enough to represent whether each property does or does-not apply to each outcome, there is no way to translate the verbal assertions into mathematical relations.

I agree with the conclusion that Pr(Box 1 is selected given the revealed coin is gold) = Pr(Box 1 is selected), however I don't understand what principle you would use to justify that. The general idea that Pr(A) = Pr(A|B) is clearly wrong, so one would need an argument, mathematical or literary, to justify why that pattern applies to selection of Box 1 in the Bertram Box problem.

One can make a literary argument that sounds like physics - e.g. "Once the box is selected, nothing that is done to examine it can change which box it is". I like that argument! However, as a generality, it often fails to account for the numerical behavior of probabilities.

There being only a 1/3 chance that the X value box was selected, the second coin has only a 1/3 chance of being different, and therefore must have a 2/3 chance of being the same.

In that fragment of your argument, you are reasoning about outcomes that involve properties of the "second" coin. So it seems to me that if you were to explicitly describe the sample space to which your argument applies, you'd need a description of outcomes that was detailed enough to describe the properties that may or may-not apply to the first and second coin in each outcome. That sample space might not be like my sample space, but it would be more detailed than simply having 3 outcomes, each defined only by which box is selected.

As I see it, you are formulating an argument that explains why a simple 3-element sample space can answer the question by implicitly discussing another sample space that has more detail.

 

[sysprog]

When discussing a problem (given in words) where each outcome may have certain properties, I think the sample space should be detailed enough to exhibit whether each outcome does or does not have each property. If this is not done then the issues of mathematics and the issues of literary interpretation get jumbled together. People make verbal assertions about phenomena. These assertions refer to properties of outcomes. Unless the sample space is detailed enough to represent whether each property does or does-not apply to each outcome, there is no way to translate the verbal assertions into mathematical relations.

I agree with the conclusion that Pr(Box 1 is selected given the revealed coin is gold) = Pr(Box 1 is selected), however I don't understand what principle you would use to justify that. The general idea that Pr(A) = Pr(A|B) is clearly wrong, so one would need an argument, mathematical or literary, to justify why that pattern applies to selection of Box 1 in the Bertram Box problem.

One can make a literary argument that sounds like physics - e.g. "Once the box is selected, nothing that is done to examine it can change which box it is". I like that argument! However, as a generality, it often fails to account for the numerical behavior of probabilities.

From my post:

If and only if the X value box is the selected box, the second coin can and must be different from the first. There being only a 1/3 chance that the X value box was selected, the second coin has only a 1/3 chance of being different, and therefore must have a 2/3 chance of being the same. Consequently, if the revealed coin is gold, the chance of the other coin in that box also being gold is 2/3.​

And from your post:

There being only a 1/3 chance that the X value box was selected, the second coin has only a 1/3 chance of being different, and therefore must have a 2/3 chance of being the same. Consequently, if the revealed coin is gold, the chance of the other coin in that box also being gold is 2/3.

In that fragment of your argument, you are reasoning about outcomes that involve properties of the "second" coin. So it seems to me that if you were to explicitly describe the sample space to which your argument applies, you'd need a description of outcomes that was detailed enough to describe the properties that may or may-not apply to the first and second coin in each outcome. That sample space might not be like my sample space, but it would be more detailed than simply having 3 outcomes, each defined only by which box is selected.

The first of the three sentences in the paragraph the remaining two sentences of which you excerpted and presented as a fragment of my argument, "If and only if the X value box is the selected box, the second coin can and must be different from the first.", I think makes it clear that the only property of the second coin I am reasoning about is whether or not it is the same as the first, which is equivalent to whether it is from a Y value box or from the X value box.

As I see it, you are formulating an argument that explains why a simple 3-element sample space can answer the question by implicitly discussing another sample space that has more detail.

As I see it:

The problem statement distinguishes two classes of box by their internal constituencies, viz, one class of box that has two constituents that are of the same kind as each other, and the other class that has two constituents that are of different kinds from each other, and two types of box constituent, viz, gold or silver, and also distinguishes the two members of the first class from each other as containing either only the first constituent type or only the second constituent type, viz both coins gold or both coins silver.

[It may be imperspicuous that the classes are not named in alphabetical order, and that the types are named before the second class is named, but that is the order in which the problem statement names them.]

Given 3 boxes, one of which is selected at random, the chance that the selected box is of the first class, class Y, which class comprises two of the three boxes, is 2/3, and the chance that the selected box is of the second class, class X, which class comprises one the three boxes, is 1/3.

Although there can be counted four outcomes for the 2 coins, viz, GG, SS, GS, and SG, the first 2 outcomes are pre-comprised in the first class, Class, Y, which has two of the three members (boxes), by 2/3 probability, and the latter two outcomes are of the second class, Class X, which class comprises only one of the three members (boxes), by 1/3 probability, so that although the probability space could be stated as

Y = 2/3(GG ⊗ SS) and X = 1/3(GS ⊗ SG),

it does not, for purposes of this problem, have to be so stated, otherwise equivalently stated, because the Class Y set has two members, and the Class X set has only one member; the two distributional expressions within X are not distinct set members, as the two members within Y are; they are different poset members, and as such are merely differently ordered expressions of the same set member, and the distinction between Class Y being a two-member set, and Class X being a one-member set, albeit one analyzable as a two-member poset, is sufficient to justify writing the probability space, for purposes of the problem, either as

Y = 2/3(GG ⊗ SS) and X = 1/3 ($\neg$(GG ⊗ SS)),

or as

Y = 2/3(Y1 ⊗ Y2) and X = 1/3 ($\neg$Y),

or in some other way that does not place the two members of Class Y in the same standing as that of the two orderings of the one member of Class X.

That avoids obscuring the fact that the two members of Class Y have together as Class Y twice the probability of being the selected box as the one member of Class X has; obviously 2/3(GG ⊗ SS) is equivalent to 2/3( $\neg$GG ⊗ $\neg$SS) but not to 1/3($\neg$(GG ⊗ SS)).

We already know that Y and X are mutually exclusive, and that Y1 and Y2 are also mutually exclusive, so we know that

Y ⇒ ($\neg$X ∧ ((Y1 ⇔ $\neg$Y2) ∧ ($\neg$Y1 ⇔ Y2))),

and when a coin is revealed to be of type G (gold), we know that

$\neg$Y2 ∧ (X ⇔ $\neg$Y1),

but that only affects the type possibilities within Class Y and the type order possibilities within Class X; it doesn't change the external probability of either class.

We knew from the start that Class Y had 2/3 of the probability of containing the selected box, and that only one of the two members of that class could be the selected box, and knowing which member it can't be has no effect on whether it is one of them.

Although the revealing of a coin type eliminates one of GG or SS within Class Y, it does not change the 2/3 probability of Class Y compared to the 1/3 Class X; it does not determine which of (GG ⊗ SS) ⊗ $\neg$(GG ⊗ SS) is true, and therefore renders a change in neither the probability of Y nor that of X.

Again, if a gold coin is revealed, that changes the internal probability share within Class Y of Y2 from 1/2 to 0 and of Y1 from 1/2 to 1, and removes the internal possibility of order SG within Class X, but it does not change the external 2/3 probability of Class Y, or the external 1/3 probability of Class X, containing the selected member, so after the revealing of a gold coin from the selected box, the probability that the other coin in the same box is also gold, is 2/3.

 

[Stephen Tashi]

As I see it:

The problem statement distinguishes two classes of box by their internal constituencies, viz, one class of box that has two constituents that are of the same kind as each other, and the other class that has two constituents that are of different kinds from each other, and two types of box constituent, viz, gold or silver, and also distinguishes the two members of the first class from each other as containing either only the first constituent type or only the second constituent type, viz both coins gold or both coins silver.

...etc.

As I see it, your treatment does not clearly define a probability space whose outcomes have all the properties you mention and you use terminology such as "the internal probability share within Class Y" that has no standard definition.

I agree that it is possible to discuss situations in real life (or imagined real life) without explicitly defining a probability space. However to apply mathematical probability theory coherently, a probability space is necessary. Part of this probability space is the set of outcomes. To define a set, one must specify the elements of the set. Your style of exposition does not make the set of outcomes clear. ( It may be that you don't intend for your arguments to be based on mathematical probability theory - in which case I don't know what standard to use in judging them.)

 

[sysprog]

@Stephen Tashi In my post (#72) I mistakenly used ⊗ (tensor product) where I meant ⊕ (XOR) -- I temporarily forgot that the encircled plus sign was in reference to modulo 2 binary addition, and didn't recall that the encircled X referred to matrix or tensor multiplication, and not to eXclusive, as the X in XOR does -- the edit time window is closed now, so I can't edit the correction into the post.