[Brain Teaser] Does this improve your chances?

[Surlethe]

You have been caught in ancient Rome and thrown into the Coliseum for pissing off the Emperor. It has been decided that your fate will be left up to fortune - will she spurn you or spread her legs for you? In the Coliseum are three doors. Behind two are hungry lions; behind the other is a sexy slave. You are to choose a door to open. If you open one of the lions', it will bound out and devour you alive. If you open the slave's, you will be freed and she will be given to you.

You make your decision and step forward. As you are about to turn the knob, the Emperor stands and the Coliseum hushes. He has had a moment of mercy, and decrees that one of the lions be led away and the door opened. It was not the door you're about to choose from. Now you have two doors to choose from instead of three.

The question before you: should you change your guess?

Post your answer & reasoning in spoilers so that you don't spoil it.

[Singular Intellect]

Yes, I change my guess.
Spoiler

Because initially I had a one in three chance of being correct. With the removal of one of the lions, my chances have been changed to one in two instead. If I don't change my guess, my initial guess will still have been based upon a one in three chance of being correct. Damn right I'm going to change my guess with the better odds.

[Redleader34]

Spoiler

Due to the paradox of math, you don't change your guess since you now have a larger chance of being right, its the 3 challenge with the goats and a car. If you picked right (you did) you have what, a 2/3ds chance of being right rather than the 1/3rd of being wrong? Also singular you may have fallen into the trap of odds changing when the events are told to you, they don't if I remember school classes.

[Grandmaster Jogurt]

Spoiler

Unless I misread something, this is just a Monty Hall problem. In which case, yes, do change, because it doubles your odds of success.

Originally, the odds are 1/3 for obvious reasons (three doors, one not-death one). If you don't change your answer, this doesn't change, either. However, when you pick originally, if you chose the safe door, if you change, you'll die. But if you chose a lion door, the other lion door is now not an option, so switching automatically picks the safe door. Making the change after the moment of mercy effectively swaps the odds, so picking a lion originally, with 2/3 odds, is the safe choice.

[Morilore]

Spoiler

Yes.

There is a 67% chance that the door you initially were going to open contains a lion. Regardless of whether the door you were about to open was correct, the Emperor wasn't going to open that one, so there is still a 67% chance that it contains a lion. (If you chose Lion 1, the Emperor must have released Lion 2; if you chose Lion 2, he must have released Lion 1; if you chose Slave, he released either Lion 1 or Lion 2.) However, because the door the Emperor opened definitely did contain a lion, there is now only one lion and one slave, so the door you did not choose has a 33% chance of containing the lion.

[General Zod]

Spoiler

I keep my choice the same. Roman leaders have generally been known to be dickish, so removing the lion may have been an attempt at instilling false confidence in me and get me to really choose the door with the lion. Thus with the increased odds of success and knowledge that it was most likely a ploy by the emperor to be a dick, I go with my original door. Even though there's probably a lion behind it anyway and the sexy slave was really just a ruse, I say why give the emperor the satisfaction of dancing to his tune?

[Rye]

Spoiler

I've heard of this problem before and the answer is (from my mathematical superiors) apparently to change your guess. I would do that out of faith in them (or doubt in myself) if such a situation presents itself.

I'll be honest, I don't get it.

The first time you choose, you have a 33% chance of being right. All's well and good.

The second time you choose, you have a 50% chance of being right.

Why isn't this true of the "stick with" choice? Both doors have 50% chance of being right, because one door's been removed and one lion has been lead away regardless. If I change my choice, it's not going to have more chance of being right than sticking with it, because the fact it was 33% before the other door was removed as an option has no impact on the current choice, which is 50/50.

[Zablorg]

Silly Surlethe, this is an old one!
Spoiler

You change your answer. It's a two out of three chance that you chose a lion first time, and so changing the answer now that the odds have changed in your favour is a good move

[Singular Intellect]

Reply to Rye:
Spoiler

I'll be honest, I don't get it.

The first time you choose, you have a 33% chance of being right. All's well and good.

And you have a 67% chance of being wrong. Meaning the odds favor you chosing one of the lions.

The second time you choose, you have a 50% chance of being right.

Correct, whereas previously it was much more likely you chose a lion, the remainng choice is gives you better odds at picking the girl.

Why isn't this true of the "stick with" choice?

Because previously you had a 67% chance of being wrong; it's better to go with 50% chance of being wrong than 67%, don't you think?

Both doors have 50% chance of being right, because one door's been removed and one lion has been lead away regardless. If I change my choice, it's not going to have more chance of being right than sticking with it, because the fact it was 33% before the other door was removed as an option has no impact on the current choice, which is 50/50.

It's because you're analysing the existing two doors without accounting for the previous odds where you were wrong with 67% certainty.

[Nephtys]

Spoiler

This is only right if you play word games. You never had a 1 in 3 choice. You've always only had a 1 in 2 choice.

It's like saying 'I'm thinking of between one and ten. Pick one. Got it? Okay, my number is odd. Assuming you didn't already guess an odd choice right now, want to reguess?'.

[Junghalli]

Spoiler

Like Rye, I'm confused. Before I had a 1/3 chance of being right. Now I have a 1/2 chance of being right. I don't see how changing my choice changes the 50/50 odds at all.

[Singular Intellect]

Reply to Nephtys:
Spoiler

This is only right if you play word games. You never had a 1 in 3 choice. You've always only had a 1 in 2 choice.

Wrong. You had a one in three choice and you made it.

It's like saying 'I'm thinking of between one and ten. Pick one. Got it? Okay, my number is odd. Assuming you didn't already guess an odd choice right now, want to reguess?'.

Let's say you have a million doors to open, whereas only one of them has a bad choice. The rest give you a million dollars. You pick one out of the a million.

Pretty good odds you you got a million dollars, right?

However, someone comes along and removes 999,998 doors except for the 'bad' one before you can open the door you picked. Now it's a fifty fifty chance.

According to you, previous odds don't matter, because it was 'always a fifty fifty chance'. Even though when you actually picked your door, the odds were a million to one you were wrong.

[Pint0 Xtreme]

I chose the correct answer but mostly because I saw this problem at the beginning of the movie, 21.

[Grandmaster Jogurt]

Spoiler

The odds aren't 50/50 in any situations. A simple way to think of this is that there are three equally-likely situations, each with two choices:

You chose lion A
----You choose to switch
----You choose to stay

You chose lion B
----You choose to switch
----You choose to stay

You chose the slave
----You choose to switch
----You choose to stay


Now examine each of the situations and its choices, and look at what it leads to. Red is death, green is victory:

You chose lion A
----You choose to switch
----You choose to stay

You chose lion B
----You choose to switch
----You choose to stay

You chose the slave
----You choose to switch
----You choose to stay


Now if you examine just the choices, "switch" is only half as likely to lead to death as "stay" is, since each of the three situations are equally likely.

[Adrian Laguna]

I stick with my choice.
Spoiler

The Emperor wants me dead. Therefore if the door I picked contained a lion, he would have just let me open it. Reducing the chances by leading a lion away is just a ruse to get me to switch and meet my doom.

[Formless]

I love this one. Its a good way to show just how hard it is for humans to understand statistics.

To those of you who know the right answer but don't believe it, you can prove it experimentally using cards and a second person to simulate the doors and the emperor. Repeat it over several trials and record the results. I think you will be pleasantly surprised.
Spoiler

You get better odds by switching, if you hadn't guessed.

[Singular Intellect]

Whoops, yeah, I screwed up by mentioning you ever had a 50% chance. My bad.

[General Zod]

I love this one. Its a good way to show just how hard it is for humans to understand statistics.

To those of you who know the right answer but don't believe it, you can prove it experimentally using cards and a second person to simulate the doors and the emperor. Repeat it over several trials and record the results. I think you will be pleasantly surprised.
Spoiler

You get better odds by switching, if you hadn't guessed.

Spoiler

I realize how the statistics involved work to a degree, and that's what this was about, but I figure hey, Roman Emperor, known for being massive dicks, I'm rolling with it and giving the Emperor a big "fuck you, I'm not dancing to your tune" spiel.

[Formless]

Spoiler

Yeah, there is the social consideration "is the tester lying," but if he is, you are doomed anyway. Personally, I'd take that risk.

[Kuroneko]

Spoiler

The question is not mathematically well-defined. We don't know the Emperor's method. It's true that the most likely intended interpretation is thus:
0) The Emperor reveals a lion, but we know nothing about why he chose to do so. This situation is completely ill-defined--Mr. Laguna's concerns being just one possible scenario.
1) The Emperor reveals a random lion that's not behind your door, and would do this regardless of the status of the the initial pick. Then a policy of always switching gives probability of winning 2/3, whereas always keeping gives probability 1/3.
2) The Emperor reveals a random lion, which just happens not to be the player's. Then it doesn't matter whether your keep or switch--your chances of winning are 1/2.
Note that the second case is perfectly consistent with the OP, especially since we have no information that the Emperor himself knows anything at all about what's behind the doors (his order could have been for workers behind the doors).

Notice that even without guessing sinister motivations for the Emperor's actions, the question is not well-defined. To emphasize, the critical difference is between:
(1) The lion was not the player's first pick, and could not have been, and
(2) The lion was not the player's first pick, but could have been.

Spoiler

This is only right if you play word games. You never had a 1 in 3 choice. You've always only had a 1 in 2 choice.

That's not true. If the experiment is repeated, if your policy is to always ignore the new information of the door, then you'll be right only 1/3 of the time.

It's like saying 'I'm thinking of between one and ten. Pick one. Got it? Okay, my number is odd. Assuming you didn't already guess an odd choice right now, want to reguess?'.

It's not comparable. The key difference is that the Emperor not only knows the correct answer, but his information never outright contradicts your first guess. If the Emperor always reveals a random lion, which could be your first pick, then you'd be right in that your chances of winning are 1/2. However, instead the Emperor always reveals a lion that wasn't the player's pick.

Suppose the Emperor doesn't know and you're always allowed to pick a revealed slave. There are two cases:
(a) If the Emperor's method is to pick a door that's not yours: it doesn't matter what you do on a revealed lion; your chances are 2/3 regardless of keep or switch in that case.
(b) If the Emperor picks a random door that could be yours: always switch. Overall probability of winning by keeping in the case of a revealed lion is 5/9, but 2/3 by switching.

[Dave]

My understanding of logic is limited to a single semester in college, and my higher mathematical ability is atrocious, but I will attempt to answer. Spoiler

Does this improve your chances? Yes and no.
Statistically speaking, your chances of survival have improved from 0.33 (repeating) to 0.5
However, this does not assist in your decision-making. Each door represented equal odds of death/survival both before and after the decree was made.

Should you change your guess?
Again, speaking only in terms of statistics, it is irrelevant whether or not you change your guess.
As to whether or not the timing of the decree is a clue as to what is behind the door you are about to open... that is a field of human relations that I am completely unfit to answer. My guess: No, continue with the door you were about to open.

[Singular Intellect]

Damnit, how to explain this concept to people?
Spoiler

For those confused, change the number of doors to a million, and the number of removed doors/lions to 999,998.

You have to pick a door out of a million. The chances are a million to one you get the right answer.

Before opening your choice, you're well aware that you have a million to one odds against you for being eaten by a lion.

But before doing so, suddenly every 999,998 doors diappear, and it's declared that between the two left over, one of them is the bad one.

In order to justify not immediately picking the other one, you're actually insisting that out of a mllion to one odds on picking the right door, you did.

This scenario is exactly the same, it merely uses a much smaller number that confuses people.

Does this help understanding what's going on statistically speaking? You're literally reversing the statistical probability to your favor. Initially you have a million to one odds against you, and then you have a million to one odds for you. We're simply dealing with smaller probability percent points in this case.

[General Zod]

Damnit, how to explain this concept to people?
Spoiler

For those confused, change the number of doors to a million, and the number of removed doors/lions to 999,998.

You have to pick a door out of a million. The chances are a million to one you get the right answer.

Before opening your choice, you're well aware that you have a million to one odds against you for being eaten by a lion.

But before doing so, suddenly every 999,998 doors diappear, and it's declared that between the two left over, one of them is the bad one.

In order to justify not immediately picking the other one, you're actually insisting that out of a mllion to one odds on picking the right door, you did.

This scenario is exactly the same, it merely uses a much smaller number that confuses people.

Does this help understanding what's going on statistically speaking? You're literally reversing the statistical probability to your favor. Initially you have a million to one odds against you, and then you have a million to one odds for you. We're simply dealing with smaller probability percent points in this case.

Spoiler

That's what I figured, but by the same token there's always the chance that the Emperor's just fucking with you and there really isn't a sexy concubine waiting for your amusement if you pick the right one. I mean this is a guy that enjoys watching people getting eaten by lions, so hey, he might just be fucking with me some more to watch the plebe dance. Why give him the additional satisfaction?

[Singular Intellect]

Spoiler

That's what I figured, but by the same token there's always the chance that the Emperor's just fucking with you and there really isn't a sexy concubine waiting for your amusement if you pick the right one. I mean this is a guy that enjoys watching people getting eaten by lions, so hey, he might just be fucking with me some more to watch the plebe dance. Why give him the additional satisfaction?

Well, presumeably the OP was serious when he said the idea was "your fate is left to fortune", not human intent. I took this to mean it was a mathematical test, not a human psychology one.

[General Zod]

Spoiler

That's what I figured, but by the same token there's always the chance that the Emperor's just fucking with you and there really isn't a sexy concubine waiting for your amusement if you pick the right one. I mean this is a guy that enjoys watching people getting eaten by lions, so hey, he might just be fucking with me some more to watch the plebe dance. Why give him the additional satisfaction?

Well, presumeably the OP was serious when he said the idea was "your fate is left to fortune", not human intent. I took this to mean it was a mathematical test, not a human psychology one.

Spoiler

True, but I find that fate generally hates me in real life with all the bizarre shit that I seem to stumble on when I'm just minding my own business. So why take the chance?