Monty Hall, Deal or No Deal and the Prisoners: An Improbable Journey
Or: The Fall of Monty Hall
Deal or No Deal is a game show featuring a case game, where the player is presented with 26 cases, each of which is randomly assigned a certain amount of money. The monetary amounts are known; it’s just not known in which case each amount resides. The player picks or is assigned one of the boxes, and then repeatedly chooses other boxes to be removed. At certain points, the player gets offered a deal: an amount of money she can get for stopping the game. If she plays till the end, the player gets whatever amount of money the last case contains.
So one particularly interesting game ended with the contestant ending up with two boxes: one containing $1, and one containing $1,000,000. The bank offered him $416,000 to stop the game right there. He said No Deal, chose the wrong case and ended up with $1.
I don’t want to get in the rationality of this choice too much. You could say the expected value of “No deal” was 0.5 * $1 + 0.5 * $1,000,000 = $500,000.50, which is more than the $416,000 for “Deal”. You could also argue that money has diminishing marginal utility: that e.g. each extra dollar you get is worth a little less to you than the previous one. In terms of life changing power, the difference between $416,000 and $1 might be a lot more than the difference between $1,000,000 and $416,000. But it depends, not in the least on one’s personal finances.
A more interesting discussion happened online: some people argued that the contestant should have switched to the other case (the one he didn’t pick at the beginning), as that would supposedly have given him a 25/26 probability of winning the $1,000,000. After all — so goes the argument — at the start of the game, he had a 25/26 probability of choosing a case not containing that $1,000,000. Now that he removed the other 24 cases, all that probability mass “went” into the 1 remaining case. This reasoning is understandable when you consider the following problem.
You’re participating in a game show. The host — call him Monty — shows you three closed doors. Monty tells you that there is $1,000,000 behind exactly one of the doors. The other doors have nothing behind them. You can pick one door, and if it leads to the $1,000,000, you get to keep the money.
Having no idea where the money is, you pick door 1. Monty then says: “I’m now going to open a door with no money behind it — and not the one you originally picked.” He opens door 3, and it turns out Monty was right: there is no money behind it.
Monty then asks you: “Now that you know there is no money behind door 3, would you like to switch to door 2?” Is it in your best interest to switch?
This problem is known as Monty Hall, and the answer is yes. In fact, if you switch, you have a 2/3 chance of winning the $1,000,000. That might be counterintuitive: there are 2 closed doors left, so shouldn’t the probability be 1/2?
But you see, you originally had a 1/3 probability of picking the door with the $1,000,000 behind it. That means there was a 2/3 probability the money was behind another door. Those probabilities haven’t changed after Monty opened door 3: there’s still a 2/3 probability the money is behind another door, but now, there’s only one other door left.
Not convinced? Let’s look at this in a more detailed way. From the start, there are three possible paths this game can take:
- You picked the door with the money behind it
- You picked one of the doors with no money behind it
- You picked the other door with no money behind it
If you picked the door with the money behind it, Monty opens one of the other doors (it doesn’t matter which). Switching then results in you picking a wrong door. So that’s one out of three paths where switching is bad.
If you picked one of the doors with no money behind it, Monty opens the other door with no money behind it. The only door left is the one with the $1,000,000, so switching gets you the money. The same is true if you picked the other door with no money behind it. So that makes two out of three paths where switching is good.
The key here is that Monty knows where the money is. No matter which door you pick at the start, he always opens another door with no money behind it.
In the drawing above, I assume the money is behind door 1 (the door I drew darker for convenience), but that doesn’t matter for the analysis: things progress in a similar way if the money is behind door 2 or 3. Also note that, if you happen to pick the door with the money at the start, Monty has a choice: he can open any of the two other doors. Which door he choses to open has no impact on the analysis (switching loses anyway in that path), so I only drew one choice. The result is that switching to the other door gives you the money in 2 out of 3 cases.
And so the 25/26 people argue that the Deal or No Deal situation above is like Monty Hall — where the contestant removing the 24 cases is like Monty opening the door — but it isn’t.
Let’s discuss the critical point more: Monty knows where the money is, and is guaranteed to open a door with no money behind it. In Deal or No Deal, the contestant could have opened any case.
In a variant of Monty Hall — hilariously named Monty Fall — Monty opens a door at random, and the result is completely different.
In Monty Fall, the setting is the same as before, except now, Monty walks across the stage, slips on a banana peel, and falls against one of the doors, accidentally opening it. “Oh, whoops”, he says. “Well, that was an accident. But as you can see, there is no money behind this door. Would you like to switch to the other closed door, now that you know this?”
Monty’s action is now completely independent of where the money is. To analyze this in detail, call the door with the $1,000,000 behind it Money Door and the doors with no money behind it Lame Door 1 and Lame Door 2. Then the situation is like this:
As you can see, there are 9 equally probable scenarios. “But”, you say, “We know Monty didn’t open Money Door! And he didn’t open the door I originally picked either!”
You’re right! Of the 9 scenarios, only 4 can actually have happened, given that you see Monty open a Lame Door which you didn’t open:
Since these 4 scenarios are equally likely, they all have a 1/4 probability. As before, switching only wins if you originally opened Lame Door 1 or 2. So switching wins with a probability of 1/4 + 1/4 = 1/2. Staying with your original choice also has a 1/2 probability of winning the money, so switching is useless in this case! And the same is true for the unfortunate contestant in the Deal or No Deal situation above. If the Deal or No Deal host knew which case contained which amount of money, and — based on that knowledge — suggested which cases the contestant should remove, then the game would be like an extended Monty Hall. But, of course, Deal or No Deal was probably not invented to make the contestant rich.
The difference with Monty Hall is that in Monty Fall, Monty doesn’t purposefully open a Lame Door you didn’t open. If we analyze Monty Hall like we did Monty Fall, we get the following:
And we can see switching wins with probability 1/3 + 1/3 = 2/3.
Keep the structure of the Monty Hall problem in mind, and consider the following problem:
Three prisoners, John, Mark and Neil, are sentenced to death. The governor has decided to let one of them go free, and has chosen the lucky prisoner at random. John, Mark and Neil each thus have a 1/3 probability of going free. Chris, the prison warden, knows which prisoner is the lucky one, but has to keep this information secret.
John is looking for some evidence that he might be the one to go free. He asks Chris the following: “Please tell me the name of one of the unlucky prisoners. If Mark is to be set free, tell me Neil’s name. If Neil is to be set free, tell me Mark’s name. If I’m to be set free, secretly flip a fair coin: if it’s heads, tell me Mark’s name, and if it’s tails, tell me Neil’s name.
Chris agrees, goes to a separate room and comes back to tell John Mark is to be executed. Relieved, John goes to Neil, and says: “I just learned Mark is going to be executed. Since it’s now between the 2 of us, this has raised my probability of going free to 1/2!” “No, it hasn’t.” Neil responds, “you still have a probability of 1/3. I, however, have a 2/3 probability of going free, so thanks for telling me!”
Who is correct here: John or Neil?
John’s reasoning may sound, eh, reasonable on first inspection, but unfortunately, he set up his experiment all wrong. One can see why with one simple observation:
John learned nothing by hearing Chris say “Mark”. There is no possible circumstance under which Chris would have said “John” instead.
Think about it. If John was the lucky prisoner, Chris would have said “Mark” or “John”. If Mark was the lucky prisoner, Chris would have said “Neil”, and if it was Neil instead, Chris would have said Mark’s name. John is relieved Chris didn’t say his name, but that makes no sense: Chris couldn’t possibly have said John’s name.
Things are different for Neil: Chris could quite probably have said “Neil” instead, and that would have sucked for Neil. So the fact that Chris didn’t say “Neil” is good news for Neil!
Analyzing this problem in a table, we get the following (I’ve only listed options that have probability greater than 0 this time):
Again, note that Chris never says “John”, so John learns nothing from his little experiment. Anyway, Chris said “Mark”, so only the following options are relevant:
As you can see, given that Chris says “Mark”, Neil is twice as likely to be the lucky prisoner than John. So the probability Neil is the lucky one is 2/3, whereas John’s probability is unchanged at 1/3.
Note that this problem is exactly like Monty Hall. By saying “Mark”, Chris opens a Lame Door; the result is that Neil is quite likely (2/3) to be the Money Door. Unfortunately for John, he’s still most probably a Lame Door.
John could have set up the experiment in a way that, if Chris responded by saying “Mark”, John would have had a 1/2 probability of going free. I leave this exercise to the reader!
As always, thanks for reading!
