Fun with Probability

[crjmurray]

Oh boy, I love the Monty Hall Paradox. Even mathematicians have been known to get this simple exercise wrong.

I was hoping some of the creationists that like to make probability arguments would throw in here.

 

[Subduction Zone]

I was hoping some of the creationists that like to make probability arguments would throw in here.

Sadly they do not tend to reply in posts that only deal with a subject that they do not understand if it has no direct connection to the evolution/creation debate.

 

[Oafman]

Great thread. I've never come across this problem before.

It is deeply counterintuitive, but after reading the explanation at least 3 times I eventually grasped it. Like others, I found that picture on wikipedia very helpful.

 

[crjmurray]

All that math and people are convinced by a cartoon goat.....

 

[Albion]

If it's not because of statistics then what's all that math doing up there?

The mathematics are being used to quantify the possibilities of an outcome that is not purely based upon randomness? For example, we might ask what are the chances of picking door number 2 if it's red? In that case, we'd be creating odds on the influence of visuals upon the contestant's thinking.

 

[crjmurray]

The mathematics are being used to quantify the possibilities of an outcome that is not purely based upon randomness? For example, we might ask what are the chances of picking door number 2 if it's red? In that case, we'd be creating odds on the influence of visuals upon the contestant's thinking.

By all means, go on.

 

[Albion]

By all means, go on.

That's it. I was asking if this might not be what the whole question posed by this thread is about.

 

[crjmurray]

That's it. I was asking if this might not be what the whole question posed by this thread is about.

No. It is about the probability of selecting the correct door.

 

[Albion]

No. It is about the probability of selecting the correct door.

The probability...based upon what considerations? If we quantify the likelihood of a child choosing chocolate over vanilla, it's not simply a matter of mathematics, is it? The taste or the popularity of the flavor, or the look of it, or other factors come into play. That's not the same as positing that a child is placed, blindfolded, at a table on which there is a dish of vanilla ice cream and one of chocolate ice cream and then asking what the percentages are that he'll go for the one over the other, strictly without any knowledge of which flavor is in which bowl.

 

[crjmurray]

The probability...based upon what considerations? If we quantify the likelihood of a child choosing chocolate over vanilla, it's not simply a matter of mathematics, is it? The taste or the popularity of the flavor, or the look of it, or other factors come into play. That's not the same as positing that a child is placed at a table on which there is a dish of vanilla ice cream and one of chocolate ice cream and then asking what the percentages are that he'll go for the one over the other, strictly without any knowledge of which flavor is in which bowl.

Then based on the OP which factors do you think are unquantifiable?

 

[Albion]

Then based on the OP which factors do you think are unquantifiable?

I was merely asking for a clarification of the meaning of the OP in that regard.

 

[Architeuthus]

Since we have so many experts on probability here, I thought we could play around with some puzzles. We'll start with an oldie.

"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 No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?"

For the sceptics, there is a simulation here. Play a few games.

I played 50 games, winning 32% of the time if I stuck with my original choice, and 64% if I switched. That supports the 1/3 -> 2/3 analysis that Michael gave.

That was a good brain teaser. I wouldn't have thought it really made any difference, but as suggested I read up on the Monty Hall Problem, and indeed there is a statistical advantage to switching. How interesting. The image they put on WIKI that seemed to make it click for me was this one:

 

[crjmurray]

I was merely asking for a clarification of the meaning of the OP in that regard.

There are no factors in the OP that cannot be quantified. That would defeat the purpose of a probability problem. There's no need to calculate the possibility of the contestant picking his favorite color door because it is not a factor. Nor do we have to calculate the probability of an earthquake stopping the show short or the probability of Monty actually being a stack of ferrets in a leisure suit.

 

[Albion]

For the sceptics, there is a simulation here. Play a few games.

I played 50 games, winning 32% of the time if I stuck with my original choice, and 64% if I switched. That supports the 1/3 -> 2/3 analysis that Michael gave.

Well, I played 10 times--that seemed sufficient to see how this works--and won 63% of the time by never switching.

 

[Architeuthus]

Well, I played 10 times--that seemed sufficient to see how this works--and won 63% of the time by never switching.

No, 10 times is not quite enough to get reasonable probability estimates. I played 25 games with switching and 25 without.

 

[crjmurray]

Well, I played 10 times--that seemed sufficient to see how this works--and won 63% of the time by never switching.

Lol. If you flip a coin ten times and it comes up heads eight times, does that mean there is 80% likelihood of flipping heads? Or do you think there is a sampling error?

 

[keith99]

Lol. If you flip a coin ten times and it comes up heads eight times, does that mean there is 80% likelihood of flipping heads? Or do you think there is a sampling error?

Well since I have the setup here is a classic, but the answer I have has a twist. Not my Twist. The twist was provided by DR. Jerry Pournelle.

A coin is tossed 8 times and comes up heads all 8 times.

Which way should you bet on the next toss, or does it not matter?

 

[morningstar2651]

The only way to lose by switching is to pick the right door the first time (1/3 chance).

• Switching wins 2/3 of the time. (2/3 chance you picked the wrong door on your first guess)
• Staying wins 1/3 of the time. (1/3 chance you picked the right door on your first guess)

Assume the car is behind door A

I selected door A first

• Switching Loses
• Staying Wins

I selected door B first

• Switching Wins
• Staying Loses

I selected door C first

• Switching Wins
• Staying Loses

 

[FrumiousBandersnatch]

That's a simple and elegant way to look at it, morningstar. I think that would have convinced me immediately when I first heard this problem. The one that actually reached me first was (as previously mentioned) extending it to 1000 doors and one car, where you pick a door (1/1000) and all the other doors, except the one with the car, are opened. Guess I'm a sucker for reductio ad absurdum

 

[crjmurray]

Well since I have the setup here is a classic, but the answer I have has a twist. Not my Twist. The twist was provided by DR. Jerry Pournelle.

A coin is tossed 8 times and comes up heads all 8 times.

Which way should you bet on the next toss, or does it not matter?

Doesn't matter.