Please note that since I wrote this article, I have been persuaded that the argument it relates to is wrong (meaning that chrysics and Mathematician and irishsultan and ChalkboardCowboy were all right from the start and I should have listened to them rather than arguing with them). Fortunately, I didn't because, for me at least, this little intellectual journey has been far more interesting than it would have otherwise been.

The correct answer for the scenario as it is worded is not 1/2 but rather 1/3 (meaning that the likelihood of winning as a consequence of staying is 2/3).

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The discussion goes on between chrysics and me. Please note that this has been written with one reader in mind, so I haven't provided much in the way of context. There are, however, links below to other relevant posts which might be of assistance to other readers. Note that the comments below that I am responding to are from /r/math over at reddit.

The correct answer for the scenario as it is worded is not 1/2 but rather 1/3 (meaning that the likelihood of winning as a consequence of staying is 2/3).

---

The discussion goes on between chrysics and me. Please note that this has been written with one reader in mind, so I haven't provided much in the way of context. There are, however, links below to other relevant posts which might be of assistance to other readers. Note that the comments below that I am responding to are from /r/math over at reddit.

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First my
preceding comment as context:

(quoted) chrysics: And if we
say that there's a Red Mary game where the contestant has picked Red and White:
your argument is again correct, there are two equally likely possibilities (MAC
and MCA). The contestant has a 50% chance of winning by switching.

me (as wotpolitan): That's
been my point right from The
Reverse Monty Hall Problem, although it might be more clearly stated in Marilyn
Gets My Goat.

As I said on my blog,
everything else has been a diversion, likely due to me not making myself
absolutely crystal clear. I agree that the argument is not valid when we have a
Red Mary game in which Red Mary is not revealed, but this would then an
isotropic White Ava or Green Ava game.

I appreciate your analysis of
why, overall, the likelihood of winning from switching across

**multiple iterations**, is 1/3 (or 2/3 in the classic Monty Hall Problem). However, ChalkboardCowboy argues that this is in contravention of the Law of Large Numbers. Would you have a response to that? (Note, my suspicion is that ChalkboardCowboy is misapplying the Law of Large Numbers.)
PS: I know that we have been
at this for a long time, but could you please take another look at the original Reverse
Monty Hall Problem article and see if I somehow failed to describe
what I meant to describe.

---

And then
chrysics’ post that I am responding to:

Perhaps my last comment was
unclear. I am not saying that the whole of your argument is valid, and the
contestant will deduce that they have a 50% chance of winning by switching. It
is emphatically

**not**the case that the contestant has a 50% chance of winning by switching, if there are no further qualifiers applied to that statement. Your argument about equally likely options existing, and thus providing a 50% chance of winning, is valid only in specific circumstances (the contestant selected Red and Green; the contestant selected Red and White), and only to somebody with knowledge that is unavailable to the contestant (the location of Mary, even if she is not revealed).
There is also an additional
scenario in which the argument does not apply. Your definition of a Red Mary
game as being

*any game in which the Red Door holds Mary*puts no constraints on which doors are selected by the contestant. As such, it requires that you consider the additional scenario (the contestant does not pick the Red Door at all. They can never win by switching, as that gets them the Red Door which by definition holds Mary) as equally likely to each of the others. The probability of winning by switching, given that Mary is behind the Red Door, is thus reduced from 50% to 1/3.
What I'm saying is that if
there exists an outside observer who knows:

1. The contestant's choice of doors, and

2. The location of Mary (assumed to be red, for the sake of simplicity)

then that observer will, in

*some*but

*not all*scenarios, deduce that the contestant has a 50% chance of winning. In another scenario, that observer will deduce that the contestant has a 0% chance of winning by switching.

The contestant themselves can
never (correctly) deduce that switching provides them with a 50% chance of
winning. They do not have the same information available to them. The
contestant is asked, once the door is opened and they find themselves in a
Revealed Red Mary (or a Revealed Green/White Ava, as the case may be), to
evaluate the probability that they win by switching. To do that, they must
weigh up all of the possible ways in which they could potentially arrive at the
scenario they now find themselves in, and must also consider how likely each of
those ways is to actually produce the scenario they find themselves in. In
doing so, they follow a procedure equivalent to that I outlined in my
earlier post.

Your argument is valid only in
the scenario that the Red Door is one of the contestant's two chosen doors.
Your argument does

*not*show that the contestant has a 50% chance of winning a Red Mary game. It shows that the contestant has a 50% chance of winning a Red Mary game*if and only if they have selected the Red Door as one of their two*. Which is true only 2/3 of the time.
The analysis I'm providing -
let me make it very clear, once again - is entirely independent of how many
times you play the game. It is not an analysis exclusively of the probabilities
you get when playing repeatedly. Nor is it an analysis

*exclusively*of the probabilities when playing a single iteration,**but it can be treated as such if that is what interests you**. Because the probabilities deduced by the contestant are in no way dependent upon how many times the game is played.
My response to ChalkboardCowboy's
statement would be that ChalkboardCowboy is exactly right. I'd be interested to
know why you think the law of large numbers cannot be applied here (or why you
think it applies but in a different way than ChalkboardCowboy says, if that's
the case).

(quoted) wotpolitan: could you
please take another look at the original Reverse Monty Hall Problem article and
see if I somehow failed to describe what I meant to describe.

I believe you've described
exactly the same process throughout, with the exception of the
one time you said that the host is committed to open the Red Door if it holds
Mary. I'll accept this was an error as you've otherwise been consistent
both before and after in saying the host will choose randomly if both selected
doors hold goats. Your initial blog post seems perfectly clear to me. You ask
for the likelihood, as deduced by the contestant, that switching will win the
car. There's not really any room for ambiguity here. We can narrow it down to,
for example, only considering the possibilities in which Mary is behind the Red
Door, and perform the analysis that way, and this does not affect the result as
it is one of many equally likely scenarios which each produce the same set of
probabilities.

First
off, I have to reiterate my rather strange sounding claim that, overall, with The
Reverse Monty Hall Problem, the contestant will benefit from a policy of staying,
winning 2/3 of the time, if the game is played repeatedly. I’ve not denied that.

Secondly,
I’ll quote myself from The
Reverse Monty Hall Problem (I keep linking to it because my original words
are there and I think that many people are responding to what they think I said
and not to what I actually said, I don’t feel particularly responsible for arguing for what
they think I said):

One day you decide to go out
to buy a new puzzle book at the massive Honty Mall. When you enter,
however, you are confronted by three doors and a rather dishevelled amateur
philosopher who swiftly talks you into trying out a variation of an old game
show puzzle (you obviously like puzzles, so it was an easy task).

The puzzle is put to you as
briefly yet comprehensively as possible:

·
There are three doors, there is a goat behind
two of the doors and behind the third is a car.

·
If, at the end of the game, you open the door
with the car behind it, you win the car.

·
First, you select

**doors (not the one door of the Classic Monty Hall Problem).***two*
·
The philosopher will then open one of the doors
you selected, revealing a goat.

·
You then have the option to switch from your
remaining selected door or stay.

·
Before being allowed to open a door, you

**provide the likelihood that a switch will win you the car (even if you choose to stay).***must*
·
The placement of the goats and car is
randomised.

Do you switch or stay, and
what is the likelihood of winning from a switch?

In the
exact situation in which the contestant finds herself, two doors

**been selected (later described in Marilyn Gets My Goat as Red-White, Red-Green or White-Green, and specified as Red-Green) and one door has been opened (specified as the Red Door in Marilyn Gets My Goat). While I do discuss the specific example of Mary revealed behind the Red Door (referred to as “Red Mary”) after the Red and Green Doors were selected, if I have correctly understood the term this situation is isomorphic with:***have*
Red Mary & Red-White

White Mary & Red-White

White Mary & White-Green

Green Mary & Red-Green

Green Mary & White-Green

Red Ava & Red-White

Red Ava & Red-Green

White Ava & Red-White

White Ava & White-Green

Green Ava & Red-Green

Green Ava & White-Green

These
constitute

**of the situations in which the contestant might find herself after selecting two doors and after the host has opened a door to reveal a goat in a single iteration, one shot instance of the Reverse Monty Hall Problem.***all*
You (chrysics)
said:

(neopolitan's) argument about equally
likely options existing, and thus providing a 50% chance of winning, is valid
only in specific circumstances (the contestant selected Red and Green; the
contestant selected Red and White), and only to somebody with knowledge that is
unavailable to the contestant (the location of Mary, even if she is not
revealed).

This sort
of misses the point, but you seemed to have got the point earlier when you
wrote:

If we say that there's a Red
Mary game where the contestant has picked Red and Green:

your argument is correct, there are two possibilities (MAC and MCA), each of which is equally likely. The contestant has a 50% chance of winning by switching.

Because
all the situations in which the contestant finds herself after the door has
been opened are isomorphic, no matter which doors she selected and no matter
which door was opened to reveal which goat, she will have a 50% chance of
winning by switching.

With
regard to the objections of ChalkboardCowboy and the Law of Large Numbers,
this law applies to “the result of performing the same experiment a large
number of times”. Remember that in the
situation in which the contestant must assess the likelihood of winning as a
result of switching, she has picked two doors, one door has opened and one goat
has been revealed. This opened door and
revealed goat tells the contestant which particular subset of mini-games the
mini-game that she is playing belongs to (in our example, MAC or MCA). If we don’t care to distinguish between
goats, then simply the opened door tells us (in our example ggC or gCg).

If we
try to apply the Law of Large Numbers the way that ChalkboardCowboy is
implying, then we don’t get to know which subset of mini-games apply with each
iteration, we just know that it is one of the set [ACM, AMC, CAM, CMA, MAC, MCA]
(or [Cgg, gCg, ggC] if we don’t care about the goat’s identity). This means that we are

**“performing the same experiment”.***not*
It
would be akin to walking into a room with three (fair) gambling machines, one
with an average payout of 1/2, one with an average payout of 1/10 and one with an
average payout of 1/100, selecting one at random and expecting to get a payout
of about 1/5 from a single machine. However,
if you go in and play the 1/2 machine 1,000 times, then you will get close to an
average payout of about 1/2. Do it a
million times and you’ll be even closer to 1/2. (This example is not entirely analogous to
what is going on with Reverse Monty Hall Problem, I know that. I am just highlighting the fact that the Law
of Large Numbers won’t work if you are playing different games.)

So,
the question I have is: did I somehow fail to make perfectly clear that I was
talking about a decision made by the contestant after the door was opened (and
hence after doors were selected) – a situation which, by your calculation,
gives the contestant a 1/2 likelihood of winning as a consequence of switching?

And a
further question is: how was my scenario substantively different to the original
question raised by Craig F. Whittaker?