Wow. We're talking about this exact same scenario in philosphy class today. Jared Morante nailed it.
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A little maths riddle...
- Thread starter Hellsgamerz
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temp5897
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Blackbird said:I'm glad you got it in the end, but you're wrong, with the bum option eliminated, you have 1 out of 2 chances to win. Option 3 is out, remember?
Maybe you should give some more attention to your English.
Blackbird, this is an age old problem...our answers are not wrong at all. I still do not see how people do not see this, even after hearing the answer. I figured it out literally in ten seconds in high school without knowing the answer...
If anyone doubts the math behind it (which is quite simple!!!) then do the experiment! It will be proven to be correct with empirical first hand evidence.
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temp5897
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Just to clarify on the goat/car problem. It would only be a 50/50 chance if, after you made your selection, the doors were randomly scrambled and you were asked to pick again.
Let us extrapolate this a bit to demonstrate to the naysayers why this works.
Suppose you were given 1 million doors. Behind one of them is car. Behind 999,999 doors there is a goat. You can pick ONE door. You choose one. You know the car is behind only one door so you have a 1 in 1 million chance of getting it right. Now out of all the doors left, the host eliminates 999,998 of them leaving two doors. You can stick with your original door or switch to the one left. Now how many of you still think it's 50/50 for this situation or the previous one described with three doors?
The answer is obvious and quite simple.
Let us extrapolate this a bit to demonstrate to the naysayers why this works.
Suppose you were given 1 million doors. Behind one of them is car. Behind 999,999 doors there is a goat. You can pick ONE door. You choose one. You know the car is behind only one door so you have a 1 in 1 million chance of getting it right. Now out of all the doors left, the host eliminates 999,998 of them leaving two doors. You can stick with your original door or switch to the one left. Now how many of you still think it's 50/50 for this situation or the previous one described with three doors?
The answer is obvious and quite simple.
Non math related riddles
How far can you walk into the woods?
Halfway
A person is driving by a cemetery and asks his friend, "How many dead people are in there.
All of them
How far can you walk into the woods?
Halfway
A person is driving by a cemetery and asks his friend, "How many dead people are in there.
All of them
This is the stupidest "math quiz" I ever heard. First of all, there's a critical piece of information missing, which is that the question only makes sense IF you don't get to look at what you first picked. Otherwise you should always switch because you'd have a 100% chance of winning. Then, there's also the issue of the "independence" of the trials, and I fully disagree with the logic presented as the "solution". It relies on a hidden assumption that may or may not be true. For those of you who think you're real math (or rather, probability) wizards, see if you can identify what the assumption is, and how it affects the outcome.
This stuff is just like religion, it's all about the fundamental assumptions. That would be true of just about all the philosophy in the world, and the value systems, the belief systems, the political systems, and etc etc. People tend to make a whole lot of assumptions in their "logic" (read: math), and 99.99% of the time they don't even realize they're doing it.
This stuff is just like religion, it's all about the fundamental assumptions. That would be true of just about all the philosophy in the world, and the value systems, the belief systems, the political systems, and etc etc. People tend to make a whole lot of assumptions in their "logic" (read: math), and 99.99% of the time they don't even realize they're doing it.
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nonsqtr said:
If I understand you correctly, you are completely wrong. There are no assumptions whatsoever in the first problem. You don't get to see what's behind the doors before you make your final choice. It's a simple problem with a simple answer. As far as thinking "we" (I'm assuming you're referring to me here based on how you phrased what you said, and my many responses here about the answers) are real math and/or probability wizards...well yes, I am.
And I won't bother explaining why but trust me on this. Since you think I am making all of these silly assumptions I would like for you to point them out. This is an old and well known problem, it is a bit silly to think that you have found some huge flaw in the extremely simple nature of it. Like I said. If you do not understand the simple math behind it, do the experiment. Period.
Well, first of all, there's a difference between "math" and "probability". Math is all about sufficiency and completeness. It's a system of operations based on fundamental assumptions. So, if you're going to post a "math" quiz, it's probably a good idea to make sure the information is "complete". I'm sure the point about "you don't get to see what's behind the doors before making your final choice" may have seemed so obvious as to be trivial, but really it's a critical piece of information that changes the whole nature of the problem. So right there, there's probably an assumption underlying the problem "as presented".
"There are no assumptions whatsoever in the first problem."
I rest my case.
Let's see if anyone else can figure out what the hidden assumption is, that relates to the independence of the trials. If not, I'll post the answer (or at least, my opinion thereof).
Edit: okay, so I just remembered I'm probably going to be gone for the rest of the week, so here it is. By the way, this is a very famous problem in basic probability. It's called the "Monty Hall problem" (among other names I'm sure). My opinion is, that you got the right answer, but for the wrong reason. In order to "correctly" arrive at the answer of 2/3, you have to spin out the logic in terms of conditional probability. The statement that "there's still a 2/3 chance that the correct answer is behind one of those doors" is incorrect, generally speaking. That would be an assumption. A true statement would be, that after Monty exposes a goat, there is now a 50% chance that you have the car in your posession. But that's a different statement than saying "you now have a 50% chance of picking the car". That's the assumption relating to the independence of the trials. And that's why you have to use conditional probability to arrive at the answer "correctly". Seemingly trivial differences like that make these probability teasers interesting and educational. And, they serve to illustrate the importance of formulating the problem correctly. Trust me on that one, I make tons of money formulating problems correctly, for people who don't have the time or patience to do it themselves. Strangely enough, probability is a very precise system of thought. There's very little room for "waving one's hands over the problem", like one can easily do with so much of the math in the world. Ah well, I'm probably being a stickler. People like me are probably a big pain in the *ss for the typical loose-thinking bass player. You did say "math" though, right? If we're going to do math, then let's do it. No fair simplifying the problem for the convenience of the answer.
"There are no assumptions whatsoever in the first problem."
I rest my case.
Let's see if anyone else can figure out what the hidden assumption is, that relates to the independence of the trials. If not, I'll post the answer (or at least, my opinion thereof).
Edit: okay, so I just remembered I'm probably going to be gone for the rest of the week, so here it is. By the way, this is a very famous problem in basic probability. It's called the "Monty Hall problem" (among other names I'm sure). My opinion is, that you got the right answer, but for the wrong reason. In order to "correctly" arrive at the answer of 2/3, you have to spin out the logic in terms of conditional probability. The statement that "there's still a 2/3 chance that the correct answer is behind one of those doors" is incorrect, generally speaking. That would be an assumption. A true statement would be, that after Monty exposes a goat, there is now a 50% chance that you have the car in your posession. But that's a different statement than saying "you now have a 50% chance of picking the car". That's the assumption relating to the independence of the trials. And that's why you have to use conditional probability to arrive at the answer "correctly". Seemingly trivial differences like that make these probability teasers interesting and educational. And, they serve to illustrate the importance of formulating the problem correctly. Trust me on that one, I make tons of money formulating problems correctly, for people who don't have the time or patience to do it themselves. Strangely enough, probability is a very precise system of thought. There's very little room for "waving one's hands over the problem", like one can easily do with so much of the math in the world. Ah well, I'm probably being a stickler. People like me are probably a big pain in the *ss for the typical loose-thinking bass player. You did say "math" though, right? If we're going to do math, then let's do it. No fair simplifying the problem for the convenience of the answer.
By the way, in Europe they customarily say "maths" instead of "math," as short for mathematicS. Actually, it makes more sense. 
Brighter minds than ours have debated this, including Marilyn vos Savant. Here's a cool link, which includes a demo:
Link Removed
I think this answer offers the simplest explanation:
"After the host opens a door and you have to decide whether to stick or switch, your choice is the same as if you could stick with your door or take both of the other two doors (one of which, admittedly, has nothing behind it). So you ought to switch, since your chance of winning is better if you get to take two doors rather than just one. In fact, you're twice as likely to win if you switch."
Brighter minds than ours have debated this, including Marilyn vos Savant. Here's a cool link, which includes a demo:
Link Removed
I think this answer offers the simplest explanation:
"After the host opens a door and you have to decide whether to stick or switch, your choice is the same as if you could stick with your door or take both of the other two doors (one of which, admittedly, has nothing behind it). So you ought to switch, since your chance of winning is better if you get to take two doors rather than just one. In fact, you're twice as likely to win if you switch."
Yeah, okay. Here's a f'r instance relating to the point I was trying to make (just so you don't think I was being a complete *sshole just 'cause I had nothing better to do). The counter-example would be, quantum entanglement. Consider this situation. You have two people in a room, along with an observer. Each of the two people has a penny. Initially, the observer can see only one person at a time. Each person flips the penny a gazillion times, and the observer observes that for each person, heads comes up 50% of the time. Now remove the partition, so the observer gets to see both people at the same time. Now, the observer tells each person to repeat the experiment, one person at a time. The first person gets 50% heads again. The second person gets 50% heads again. Would you immediately jump to the conclusion that the two sets of trials are completely independent? Many people would.
But now, let's say the observer asks both people to repeat their experiments, but this time they have to do it "simultaneously" (or, let's say, one trial at a time, so the observer can still observe each one by itself). What the observer now sees, is that each person "by themself" gets 50% heads, but whenever one person gets a head, the other person gets a tail, and vice versa. Let's say that correlation occurs 100% of the time. And keep in mind that the observations are "sequential", in other words, one person flips a coin, then the other, then the other, and so on. What conclusion would you reach now? Probably that the two sets of trials aren't "fully independent", right? And that, is exactly what occurs in a quantum entanglement experiment.
I'm not saying that flipping pennies is like quantum entanglement. But I "am" saying that one must be very careful to identify one's assumptions when formulating a solution in terms of a probability model. The "math" is the more trivial part of the solution. It's the formulation of the model that must be correct. If the model is wrong, then all the best math in the world isn't going to help.
But now, let's say the observer asks both people to repeat their experiments, but this time they have to do it "simultaneously" (or, let's say, one trial at a time, so the observer can still observe each one by itself). What the observer now sees, is that each person "by themself" gets 50% heads, but whenever one person gets a head, the other person gets a tail, and vice versa. Let's say that correlation occurs 100% of the time. And keep in mind that the observations are "sequential", in other words, one person flips a coin, then the other, then the other, and so on. What conclusion would you reach now? Probably that the two sets of trials aren't "fully independent", right? And that, is exactly what occurs in a quantum entanglement experiment.
I'm not saying that flipping pennies is like quantum entanglement. But I "am" saying that one must be very careful to identify one's assumptions when formulating a solution in terms of a probability model. The "math" is the more trivial part of the solution. It's the formulation of the model that must be correct. If the model is wrong, then all the best math in the world isn't going to help.
It's the fact that the host knows which is the winner that affects the probability, and the fact that he gives you some hint as to the winner (thus changing the probability) by removing the non-winning door or doors.
If you picture the following (for the million door example)
1) You pick a door
2) Host removes 999,998, leaving 2 doors
3) Someone else comes along and picks a door without having seen what happened before
That person would have a 50% chance of picking the right door, as to them both are doors with nothing to differentiate between them.
If you picked, you'd have a *much* higher chance of winning if you picked the second door as the odds that you picked the right door in the first place (1 in a million) are far less than the odds of winning if you pick the second door (which have to be at least 50%).
If you picture the following (for the million door example)
1) You pick a door
2) Host removes 999,998, leaving 2 doors
3) Someone else comes along and picks a door without having seen what happened before
That person would have a 50% chance of picking the right door, as to them both are doors with nothing to differentiate between them.
If you picked, you'd have a *much* higher chance of winning if you picked the second door as the odds that you picked the right door in the first place (1 in a million) are far less than the odds of winning if you pick the second door (which have to be at least 50%).
Jared Morante said:
Forgive my stupidity (as usual), but if the doors were randomly scrambled and you were asked to pick again, wouldn't you be exactly where you were when you started?
Jared Morante said:Let us extrapolate this a bit to demonstrate to the naysayers why this works.
Suppose you were given 1 million doors. Behind one of them is car. Behind 999,999 doors there is a goat. You can pick ONE door. You choose one. You know the car is behind only one door so you have a 1 in 1 million chance of getting it right. Now out of all the doors left, the host eliminates 999,998 of them leaving two doors. You can stick with your original door or switch to the one left. Now how many of you still think it's 50/50 for this situation or the previous one described with three doors?
The answer is obvious and quite simple.
No, it's not 50/50 anymore, but you have changed the numbers so the odds are obviously different. Besides, why would the host eliminate 999.998 doors after the first selection is made? What kind of game allows rules to be changed as the game's being played? What's this, Calvinball?
lump said:By the way, in Europe they customarily say "maths" instead of "math," as short for mathematicS. Actually, it makes more sense.
Thanks. I'll remember that.
