Math problem

[Cosmic Wonder]

got, it, under 30 seconds, and no caps.

[Russell Letson]

I wish I could do arithmetic bare-headed.

[Cosmic Wonder]

Russell, love your avitar. Bill the cat is my hero, mostly for his well thought out philosophy and lifestyle.

Mike

[Russell Letson]

Thpppt.

[Deleted]

Some interesting factoids about the Monte Hall Dillemma...

It IS conter-intuitive in a big way. The most mail ever generated in Marilyn Vos Savants collumn was due to her (accurate) answer. Much of the record-amount of hate mail decrying her answer was from mathmaticians, computer science professionals, and scientists! The outcry was so alarming, that she had some second thoughts about ever breaching the subject again.

The problem, which stumps many math-related folks is apparently not "counterintuitive" to some professional gamblers, who immediately provided the correct answer, but were unable to explain their reasoning.

If folks like the dark art of probabilities, I'd recommend "The Drunkard's Walk". It reflects on the fact that many of the mysterious coincidences we experience (like hearing a song on the radio that you had been thinking of, or receiving a call from a long-lost friend immediately after thinking about them) are not unusual events at all, and would be remarkable only if they did not occur regularly.

[dradtke]

I got the first one in less than a minute (fewer than one minute?) as a theater and English major.

And I remember arguing with a friend in high school about the problem that came to be called Monte Hall. Similar answer, different name, that I read about. He could not be convinced he was wrong, and we got into quite an argument. Even though he was wrong, now he's a multi-millionaire and I'm still slogging along at work.

[Cornflake]

"It reflects on the fact that many of the mysterious coincidences we experience (like hearing a song on the radio that you had been thinking of, or receiving a call from a long-lost friend immediately after thinking about them) are not unusual events at all, and would be remarkable only if they did not occur regularly."

I've never heard of the book but that seems intuitively correct to me, and I've had the same thought. If a thousand strange things could happen and each has a one in a thousand chance of happening, one of them will happen with some regularity.

[omaha]

Jun 15, 2009 15:42:03 GMT -5 @dharmabum said:

If anyone likes probability issues, ever hear of the Monte Hall Dillemma? (That one usually stumps math-oriented folks as easily as the rest of us).

I was in such a hurry yesterday, I forgot you had mentioned this one. What a great "puzzle"! A total classic.

It took me quite a bit of noodling on it to get my brain wrapped around the answer. It somehow seemed so counterintuitive!

But when you get that your odds of picking the "goat" in the first place are 2/3, then it becomes clear that switching your choice after one door (and one goat) is revealed will improve your odds.

Love that stuff!

[millring]

Jun 16, 2009 14:44:28 GMT -5 @omaha said:

Jun 15, 2009 15:42:03 GMT -5 @dharmabum said:

If anyone likes probability issues, ever hear of the Monte Hall Dillemma? (That one usually stumps math-oriented folks as easily as the rest of us).

I was in such a hurry yesterday, I forgot you had mentioned this one. What a great "puzzle"! A total classic.

It took me quite a bit of noodling on it to get my brain wrapped around the answer. It somehow seemed so counterintuitive!

But when you get that your odds of picking the "goat" in the first place are 2/3, then it becomes clear that switching your choice after one door (and one goat) is revealed will improve your odds.

Love that stuff!

And I still hear it as a question full of "...well, that depends on how the rules are first defined". I guess the math escapes me. I can't get past the difference it makes why and when Monte reveals the one door.

[omaha]

Here's the way I visualize it.

You have three doors: One with a prize and two with a "goat".

You pick one. You have one chance in three of being right, which means you have two chances in three of being wrong. Which is to say, you are "probably" wrong in your first guess.

After your pick, Monte opens a door. Because he knows where the goats are, he is always going to show you a goat. Then he asks you if you want to switch your choice.

And you should and here's why: You had two chances in three of being wrong in your first guess. You now have the ability to unambiguously rectify that by switching your choice. You go from having one chance in three (your initial odds) to having two chances in three.

The only way you lose is if you picked the prize in the first place. But the odds of that are low. So you switch, and take the better odds.

[Doug]

And I don't want what Jay's got on his table
Or the box Carol Merrill points to on the floor
No, I'll hold out just as long as I am able
Until I can unlock that lucky door
Well, she's no big deal to most folks
But she's everything to me
Cause my whole world lies waiting behind door number three

[millring]

"You go from having one chance in three (your initial odds) to having two chances in three. "

That's where it loses me. I still don't understand why knowing that the door that Monte has revealed is one of the wrong ones makes my initial choice wrong if it was right in the first place (something that Monte's reveal did not disprove).

[Fingerplucked]

Jun 16, 2009 15:05:25 GMT -5 @omaha said:

Here's the way I visualize it.

You have three doors: One with a prize and two with a "goat".

You pick one. You have one chance in three of being right, which means you have two chances in three of being wrong. Which is to say, you are "probably" wrong in your first guess.

After your pick, Monte opens a door. Because he knows where the goats are, he is always going to show you a goat. Then he asks you if you want to switch your choice.

And you should and here's why: You had two chances in three of being wrong in your first guess. You now have the ability to unambiguously rectify that by switching your choice. You go from having one chance in three (your initial odds) to having two chances in three.

The only way you lose is if you picked the prize in the first place. But the odds of that are low. So you switch, and take the better odds.

The reason it works this way is because Monty will always open a door, and he's not going to reveal the prize, he always shows you a goat when he opens that door.

Wikipedia has a good visual diagram of how it works under the heading "Popular solution":

en.wikipedia.org/wiki/Monty_Hall_dilemma

[millring]

Jun 16, 2009 15:22:15 GMT -5 Fingerplucked said:

Jun 16, 2009 15:05:25 GMT -5 @omaha said:

Here's the way I visualize it.

You have three doors: One with a prize and two with a "goat".

You pick one. You have one chance in three of being right, which means you have two chances in three of being wrong. Which is to say, you are "probably" wrong in your first guess.

After your pick, Monte opens a door. Because he knows where the goats are, he is always going to show you a goat. Then he asks you if you want to switch your choice.

And you should and here's why: You had two chances in three of being wrong in your first guess. You now have the ability to unambiguously rectify that by switching your choice. You go from having one chance in three (your initial odds) to having two chances in three.

The only way you lose is if you picked the prize in the first place. But the odds of that are low. So you switch, and take the better odds.

The reason it works this way is because Monty will always open a door, and he's not going to reveal the prize, he always shows you a goat when he opens that door.

Wikipedia has a good visual diagram of how it works under the heading "Popular solution":

en.wikipedia.org/wiki/Monty_Hall_dilemma

yeah, I read through the wiki and diagrams (with the signs and arrows and stuff -a guthrie) and I still say...

That's where it loses me. I still don't understand why knowing that the door that Monte has revealed is one of the wrong ones makes my initial choice wrong if it was right in the first place (something that Monte's reveal did not disprove).

[omaha]

Jun 16, 2009 15:19:39 GMT -5 millring said:

"You go from having one chance in three (your initial odds) to having two chances in three. "

That's where it loses me. I still don't understand why knowing that the door that Monte has revealed is one of the wrong ones makes my initial choice wrong if it was right in the first place (something that Monte's reveal did not disprove).

It doesn't, and just because there is a "best" strategy, it does not follow that you can't lose. If you pick the prize with your first guess, you'll lose, even if you follow this "optimum" strategy.

Here's a quick scenario: Assume Door #1 has the prize.

You pick Door #1 : You lose (Monte reveals door #2, you switch to #3, you're hosed).
You pick Door #2 : You win (Monte reveals door #3, so you switch to #1)
You pick Door #3 : You win (Monte reveals door #2, so you switch to #1)

Bottom line: You can improve your odds from a raw 1/3 to a respectable 2/3 with this strategy.

[Fingerplucked]

Jun 16, 2009 15:25:39 GMT -5 millring said:

That's where it loses me. I still don't understand why knowing that the door that Monte has revealed is one of the wrong ones makes my initial choice wrong if it was right in the first place (something that Monte's reveal did not disprove).

Three things are constant:

1. Monty (Monte?) will always open a door.
2. Monty will only open a door showing a goat.
3. You will always change your guess.

Changing your guess, along with the first two assumptions is what gives you a 66% chance of getting it right. But it also means that if you guessed correctly the first time, you will lose, because you will still change your guess.

-------------

You probably have a mental block on this one. I know about mental blocks. I know what 12*12 is, what 11*11 is, and what 6*6 is.

I couldn't figure out T-bob's math problem in the OP. I knew that it must be something other than what I kept trying to make of it, because 6 times 6 just didn't fit the pattern I saw. And then after 20 or 30 minutes the clump of mud in my head passed and I realized that 6 times 6 is not 64.

So I sort of got the answer to his question in 5 seconds, if you don't count the 30 minutes in the middle.

[Russell Letson]

Even though this stuff makes my teeth hurt, I think the root of John's puzzlement might be that it doesn't matter whether one's first choice turns out to have been correct or not--after the first round of revelation, you still don't know where the car is, and you're offered the chance to change your choice. At that point, you can flip a coin, because the 1/3 probablity has gone to a 1/2 probablity--and you still don't know, so "right" and "wrong" are beside the point.

Am I right to be reminded of blackjack here?

[patrick]

Jun 16, 2009 15:27:36 GMT -5 @omaha said:

Jun 16, 2009 15:19:39 GMT -5 millring said:

"You go from having one chance in three (your initial odds) to having two chances in three. "

That's where it loses me. I still don't understand why knowing that the door that Monte has revealed is one of the wrong ones makes my initial choice wrong if it was right in the first place (something that Monte's reveal did not disprove).

It doesn't, and just because there is a "best" strategy, it does not follow that you can't lose. If you pick the prize with your first guess, you'll lose, even if you follow this "optimum" strategy.

Here's a quick scenario: Assume Door #1 has the prize.

You pick Door #1 : You lose (Monte reveals door #2, you switch to #3, you're hosed).
You pick Door #2 : You win (Monte reveals door #3, so you switch to #1)
You pick Door #3 : You win (Monte reveals door #2, so you switch to #1)

Bottom line: You can improve your odds from a raw 1/3 to a respectable 2/3 with this strategy.

I think the way the odds are being calculated in this game are misleading. Because of the way the game is played, your odds were always 1/2.

When you pick a door, any door, you don't get to find out if that door was the winner before the game changes. It doesn't matter which door you choose first, because they will always discard one door and then have you play again. But in that new game, you always have a 50/50 chance of picking the winner.

Imagine this scenario: I'm playing the game with Monty. He tells me to pick a door out of three. Instead, I begin doing a medley of Neil Young hits in my most obnoxious quavering falsetto voice. The producer, afraid that the audience will just run away in nausea, tells Monte to just continue and they reveal one of the losers. My game is now 1/2, without having picked a door or having to switch.

[patrick]

Obviously great minds work in the same channels.

[millring]

Alright all you smarty-pantses, you go ahead and change horses in mid-stream. But don't come crying to me when you drown. I just washed my floors and I don't want dead whiners dripping river water all over my clean linoleum.