Please note that since I wrote this article, I have been persuaded that the argument it contains is wrong. 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 following is tediously pedantic, because I have to walk people through the process step by step. Please try to read all of it and go with it. Don’t jump ahead, ask yourself each question and see if you get the same answer as Marilyn. If you don’t, then we can discuss that.
It’s pretty universal from all the comments (both on the blog and over at reddit) that I must have it wrong. I’ve tried repeatedly to explain that, also pretty universally, the criticisms are based on a misunderstanding of my scenario. That’s what this article is all about.
It’s for those people who seem to almost deliberately misunderstand what I have written.
I don’t think people deliberately misunderstand. I think that it’s because what I am talking about is so close to something else that is widely understood, most people stop really listening and presume that I am talking about the thing that they are familiar with.
There have been some people who have raised other, excellent points. I will try to address those in a later article, so that this can focus on one thing – the actual scenario of the Reverse Monty Hall Problem. I’ll do it in the form of a little role-play.
Post Burnage Comment:
I would like to stress that this scenario could be performed in real life. To do so, all we would have to do is:
- engage the services of Angelina Jolie and Marilyn vos Savant,
- engage the services of two goats, one called Mary and the other called Ava,
- establish a broadcast studio called “the Craig F. Whittaker Memorial studio”,
- purchase three doors, one Red, one White and one Green, and
- organise for the concept designer to donate the car as a prize (but we could conceivably use a different car).
Holly Mant (played by Angelina Jolie) is the host of the single episode game show Marilyn Gets My Goat, broadcast from the Craig F. Whittaker Memorial studio.
There is one single contestant, Marilyn vos Savant, who plays one single round.
Holly explains to Marilyn the key features of the game:
There are three doors:
- A Red door the left,
- A White door in the middle, and
- A Green door to the right
Holly shows Marilyn pictures of the doors:
|Note the the doors are different styles (images taken from various stock catalogues)|
Behind each of the doors is hidden one of two goats (Mary and Ava) or a car:
The placement of goats and car is completely random
Holly shows Marilyn pictures of the goats and the car (note carefully the non-interchangeability of the two goats):
Marilyn will be asked to assess various relevant probabilities during the process (so all the questions below are posed to Marilyn and the answers are Marilyn’s).
If at any time, Marilyn is uncertain about anything, she can ask any question she likes of Holly.
The game will proceed with the following steps:
Step 1. Marilyn will be asked questions about initial likelihoods
Step 2. Marilyn will select two doors at random
Step3. Marilyn will be asked questions about “post-selection” likelihoods
Step 4. Holly will open a door on instructions from her producer and will offer Marilyn
the opportunity select from the two remaining closed doors (either switching or
the opportunity select from the two remaining closed doors (either switching or
Step 5. Marilyn will be asked questions about “post-door-opening” likelihoods
Step 6. Once Marilyn provides the correct answer, she is permitted to open the door
of her choice.
of her choice.
The game commences
Q1 What is the likelihood of the car being behind the Green door?
There is an equal likelihood that the car is behind any individual door, irrespective of the colour, so the likelihood of the car being behind the Green door is 1/3
Q2 What are the possible distributions of goats and car behind the Red, White and Green doors, using the terms M for Mary, A for Ava and C for the Car?
- Red-A, White-C and Green-M
- Red-A, White-M and Green-C
- Red-C, White-A and Green-M
- Red-C, White-M and Green-A
- Red-M, White-A and Green-C
- Red-M, White-C and Green-A
Q3 How likely are each of these distributions?
They are all equally likely at 1/6
Marilyn selects her two doors at random – she selects Red and Green
Q4 What possible choices of doors did you have, Marilyn?
There were only three choices:
And these are functionally equivalent to:
- Red and White
- Red and Green
- White and Green
- Not Green
- Not White
- Not Red
Q5 Given that you selected your doors at random, what was the likelihood of each of these choices?
Each choice was equally likely, therefore 1/3
Q6 Now that you have selected Red and Green, what is the probability that you selected Red and Green?
Q7 What are the possible distributions of goats and car behind the Red and Green doors (using A for Ava and M for Mary)?
- Red-A and Green-C
- Red-A and Green-M
- Red-C and Green-A
- Red-C and Green-M
- Red-M and Green-A
- Red-M and Green-C
Q8 How likely are each of these distributions?
All equally likely at 1/6Q9 What is the likelihood that the producer will instruct Holly to open the Red door?
The likelihood of Red C is 1/3 and the likelihood of Green C is 1/3. In either of these cases the producer will instruct Holly to open the other door. Therefore we have:
There is also a 1/3 likelihood that there are goats behind both the Red door and the Green door. In this case, I don’t know the basis on which the door to be opened will be selected, so the best I can say is that it will be random.
- a 1/3 likelihood that the Red door must be opened in accordance with the rules, and
- a 1/3 likelihood that the Green door must be opened in accordance with the rules.
(Holly takes this as a clarifying question and confirms that, in these circumstances, the door will be selected at random.)
Therefore, if there are two goats behind the Red and Green doors, there is a 1/2 likelihood that the Red door will be opened and 1/2 likelihood that the Green door will be opened. Therefore we have:Q10 What is the likelihood, that the Red door would have to be opened, in accordance with the rules, on the basis that that the car was behind the Green door?
Totalling this up, we have:
- a 1/6 likelihood that the Red door is opened at random because there are goats behind both the Red door and the Green door, and
- a 1/6 likelihood that the Green door is opened at random because there are goats behind both the Red door and the Green door
- a 1/2 likelihood that the Red door will be opened, and
- a 1/2 likelihood that the Green door will be opened
There is a likelihood of 1/2 that the Red door will be opened. There is a likelihood of 1/3 that a goat is behind the Green door and a likelihood of 1/6 that there is a goat behind both doors. Using conditional probability, we can say that if the Red door is opened, then the likelihood that it was opened because there is a car behind the Green door is 2/3.
This is, effectively, equivalent to the answer to the classic Monty Hall Problem. However, I note that Holly has not yet opened the door.
Please stay with me. This is where it gets interesting!
When Craig F. Whittaker posed his question in 1990, he asked this:
“Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?”
Note that Whittaker states that the host has opened the door and that only then is the contestant offered the opportunity to switch. The contestant must calculate the likelihood of benefitting from a switch after the door has been opened.
Does it make a difference?
Holly opens the Red door revealing Mary the Goat and offers Marilyn the opportunity to switch from the Green door to the White door
Q11 What is the current likelihood that there was a goat called Mary hidden behind the Red door?
1/1Q12 Assuming that Holly has told the truth, what is the current likelihood that there is a car behind either the White door or the Green door?
1/1Q13 What are the possible distributions of goat and car behind the two remaining closed doors, knowing that Mary was behind the Red door?
- White-A and Green-C
- White-C and Green-A
Q14 How likely were each of these distributions before the goats and cars were placed?
They were equally likely distributions, out of six possible, so 1/6
Q15 How likely are each of these distributions now?
They remain equally likely distributions, but now out of two possible, so 1/2Q16 What about the likelihood that the producer would instruct Holly to open the Red door being 2/3?
That was before Holly was instructed to open the Red door. The likelihood of the Red door having been opened now is 1/1 irrespective of the motivation. Similarly:
In these two instances, we know that prior probabilities have changed as a consequence of the door being opened. The same applies to the likelihood that the producer might choose the Red door.
- the likelihood of the car being behind the Red door was 1/3. But now it is 0, and
- the likelihood of Mary being behind the door was 1/3, but now it is 1/1
Q17 So, Marilyn, in conclusion, what is likelihood of the car being behind the White door, the single door that you did not originally select?
Exactly the same as the likelihood of it being behind the Green door, 1/2
Holly offers Marilyn the opportunity to switch
Given 15 years or more of being convinced that a switch in a Monty Hall game is the best strategy, despite the logic above - or because she realises that it doesn't make a difference, Marilyn does not switch.
Holly opens the Green door
If you want to know if Marilyn got Holly’s goat, please toss a coin.
Heads she did, tails she didn’t.
As mentioned above, this argument is wrong. Precisely which bit is wrong is a little vexed. I originally thought the biggest issue nestles in Q10 and Q16. I still think it does, but many below argue that the issue is in Q15.
I have finally got around to addressing this in (My) Ignorance Behind Marilyn Gets My Goat, hopefully without sparking huge controversy this time. See also The Whole Monty Hall Debacle.