And Behind Door No. 1, a Fatal Flaw

[Aggie87]

Nope.

Switching or not switching are the only choices you have. The results/outcomes only come into play after the choice is made.

You have four possible outcomes, but they result from only two choices.

Switching or not switching ARE the only choices you have, so it IS a binary decision (either/or), which I think is what you're trying to say. But it's still a 67/33 split of likelihoods, not 50/50.

[Niko]

It (50-50) refers to what you can guarantee as far as your choice being the right one for any given time, before the results are revealed.

i tend to call that number zero - matter of personality i guess

[JSngry]

The only correct odds that the smart choice will be successful (i.e. winning the car) are 67/33, not 50/50.

Of course they are. That has never been a point of contention.

[John L]

Jim: Maybe it is the whole notion of prior probabilities that has you reeling. We know that the car IS behind one of the doors. If it is door number 1, then it is behind door number one with probability one, not 1/3 The problem is that we don't know that, and therefore have to take our chances by assessing odds and choosing a strategy that maximizes our chances of getting the car. If we have no reason to believe before hand that there is any greater probability that the car is behind door number 1, door number 2, or door number 3, then we give them odds of 1/3, 1/3, 1/3. Those are not actual probabilities, as those are 1.0,0. That is a representation of our knowledge of the likelihoods, which is nill. In this case, we would be happier if we could choose two doors and not just one, and win the car if it is behind either of the two. That would double our chances of winning. In fact, the switching strategy does just that. That is all that we are saying here. Nothing else.

Life is uncertain. We choose strategies like this all the time. Jazz musicians go to New York City because it can increase their chances of success. But, in some cases, it might not. We have to assess the odds and decide.

Edited April 11, 2008 by John L

[JSngry]

Nope.

Switching or not switching are the only choices you have. The results/outcomes only come into play after the choice is made.

You have four possible outcomes, but they result from only two choices.

Switching or not switching ARE the only choices you have, so it IS a binary decision (either/or), which I think is what you're trying to say. But it's still a 67/33 split of likelihoods, not 50/50.

I'm not trying to say it, I am saying it. And have been!

You have a 50-50 chance of getting the 67/33 odds.

[JSngry]

And since 33% of the time your 50% decision will be wrong, your odds are really only 16.5%

:g :g :g :g :g

Where was this thread on April 1?

[Dan Gould]

If the odds somehow were 50/50 for one incident, then it would be 50/50 for 1000 incidents too, since they only consist of 1000 single incidents counted together. I can't understand how you can think that it's a 50% chance for one single incident, but a 67 % chance for 100 or 1000 of them. That's just illogical.

This is where it all breaks down. I too can't understand how Jim keeps stating that he understands that the odds improve by switching yet continues to think that in a singular event the odds are somehow 50-50. If they are 67-33 over 1000 iterations, then they are 67-33 for every single iteration.

[John L]

If the odds somehow were 50/50 for one incident, then it would be 50/50 for 1000 incidents too, since they only consist of 1000 single incidents counted together. I can't understand how you can think that it's a 50% chance for one single incident, but a 67 % chance for 100 or 1000 of them. That's just illogical.

This is where it all breaks down. I too can't understand how Jim keeps stating that he understands that the odds improve by switching yet continues to think that in a singular event the odds are somehow 50-50. If they are 67-33 over 1000 iterations, then they are 67-33 for every single iteration.

There is the so-called law of large numbers in statistics that says that if you play the switching strategy enough times in a row, the the number of times that you win the car will converge to 2 out of 3. If you only play a few times, there is a strong probability that you won't get the car two out of three times. But that is something different entirely than assessing the odds of a single event.

Edited April 11, 2008 by John L

[Swinging Swede]

so you make one of two choices.

And that gives you a 50-50 chance at getting the 2 out of 3 chance.

But if you would have a 50% chance of getting a 67% chance, and a 50% chance of getting a 33% chance, then you would on average only have a 50% chance of winning the car, which just isn't the case. With the knowledge you have, you have a 67% chance of winning the car.

Just as Monty has a 100% chance by the way. Otherwise, with your way of reasoning, Monty would have a 50-50 chance of getting a 3 out of 3 chance, whatever that means.

By the way, remember the alternative case I mentioned earlier with 100 initial doors, and Monty opening 98 of them, with 2 doors remaining. Would you then say that you have a 50-50 chance at getting a 99 out of 100 chance?

[JSngry]

Jim: Maybe it is the whole notion of prior probabilities that has you reeling.

No, what has me reeling is the unspoken notion that there's no merit whatsoever in counter-logical decision making.

Because as you alluded to earlier, sometimes the mojo is working. And when it is, it behooves one to listen.

Now maybe some people don't have no mojo,in which case, I can only say, hey, too bad.

But just as it is dangerous, sometimes fatally so, to put words into the mojo's mouth, it is equally dangerous to never listen to it at all.

It's a life's work, getting in touch with the mojo is, but if the alternative is to execute preordained odds, then one cannot complain when preordained results - including living somebody else's life instead of of your own - transpire.

Me, I'm gonna switch. Why not? It's a one-off and that's the best shot, although it ain't gonna be perfect.

But if I smell a goat fart coming out from behind the curtain I'd be switching to, hey, I'm standing pat.

Does the online simulator offer goat farts?

[John L]

My God, Jim. You really do sound more like an artist than a mathematician. who would have known?

[Swinging Swede]

You have a 2 out of 3 chance of being right, but only a 50-50 chance that you'll get that 2 out of 3 chance.

Actually you have a 100% chance of getting that 2 out of 3 chance, and you realize it by switching door.

[Jazzmoose]

So what if you already have a car, but don't have a goat?

[Swinging Swede]

I still say you have a 66-33 chance at getting the 2 out of 3 chance.

Then that means you have three choices to make - switch, don't switch, and....?

A 2 out of 3 chance does NOT mean that you have three choices. In the example I mentioned with 100 initial doors, you have a 99 out of 100 chance, but that doesn't mean that you have one hundred choices. You still have only 2 doors to choose between, but by switching you have a 99% chance of winning the car.

[JSngry]

If the odds somehow were 50/50 for one incident, then it would be 50/50 for 1000 incidents too, since they only consist of 1000 single incidents counted together. I can't understand how you can think that it's a 50% chance for one single incident, but a 67 % chance for 100 or 1000 of them. That's just illogical.

This is where it all breaks down. I too can't understand how Jim keeps stating that he understands that the odds improve by switching yet continues to think that in a singular event the odds are somehow 50-50. If they are 67-33 over 1000 iterations, then they are 67-33 for every single iteration.

Dear, sweet Dan -

I am not saying that the odds of switching having a successful outcome are ever 50-50 instead of 67-33.

But let's look at this non-linearly and/or non 3-D: What are the odds that the 67-33 odds will come to pass in a one-time only scenario with only two possible outcomes? 67-33? No, that's the odds for the choice itself being successful in theory, on an endless stream of Ultimate Outcomes, not the odds for the choice actually materializing in real time at any one time.

This is not about the odds, it's about the odds of the odds. Can you guarantee that in a one shot game that the 67% outcome will be the one that materializes? No, of course not. Just as you cannot guarantee that it will not be. So even though it is more likely that one choice will be successful, there's still the nagging little inconvenience that that choice will not be answering the door this time around. And yes, there is a 1 in 3 chance that will be the case for any given time.

But will this be that time? Or not?

This is not about the odds, it's about the odds of the odds. That and goat farts...

Edited April 11, 2008 by JSngry

[JSngry]

My God, Jim. You really do sound more like an artist than a mathematician. who would have known?

I was offered math scholarships out of high school, but even then realized the limitations of literality...

[Swinging Swede]

The only way to have 50/50 occurs if YOU decide to open one of the two remaining doors (and not Monty) and that door reveals a goat.

Ok, one...more...time...

50-50 as I'm using it does not refer to the probability that switching will be successful. That is clearly 2/3.

It (50-50) refers to what you can guarantee as far as your choice being the right one for any given time, before the results are revealed.

You can never guarantee that your choice is the right one, unless you're Monty. Whichever door you pick you might lose, so the guarantee factor is 0.

[JSngry]

You have a 2 out of 3 chance of being right, but only a 50-50 chance that you'll get that 2 out of 3 chance.

Actually you have a 100% chance of getting that 2 out of 3 chance, and you realize it by switching door.

So you're saying that you do not have the option of choosing not to switch?

[JSngry]

The only way to have 50/50 occurs if YOU decide to open one of the two remaining doors (and not Monty) and that door reveals a goat.

Ok, one...more...time...

50-50 as I'm using it does not refer to the probability that switching will be successful. That is clearly 2/3.

It (50-50) refers to what you can guarantee as far as your choice being the right one for any given time, before the results are revealed.

You can never guarantee that your choice is the right one, unless you're Monty. Whichever door you pick you might lose, so the guarantee factor is 0.

No, you can guarantee that you've either chosen correctly or not.

Seems kinda...obvious, that one does...

[JSngry]

I wanna play Let's Make a Quantum Deal.

[Aggie87]

No, you can guarantee that you've either chosen correctly or not.

Seems kinda...obvious, that one does...

How do you guarantee that you've chosen correctly? I'd like to have that talent!

Edited April 11, 2008 by Aggie87

[Swinging Swede]

The only correct odds that the smart choice will be successful (i.e. winning the car) are 67/33, not 50/50.

Of course they are. That has never been a point of contention.

Well, you did actually say the following (see question in red):

So if the favorable odds are not going to apply every time and we only have one incident with which to work, what other possible odds are there than 50/50 that the smart choice will be successful this one time? Not that it should be, but that it will be?

So I answered that the other possible odds than 50/50 that the smart choice will be successful this one time are 67/33. Actually it is the only correct odds, 50/50 is not.

[Dan Gould]

Jim: Maybe it is the whole notion of prior probabilities that has you reeling.

No, what has me reeling is the unspoken notion that there's no merit whatsoever in counter-logical decision making.

No, there is merit to counter-logical decision making.

there just isn't any logic to it.

I mean, let's consider another example. The Florida lottery just started a new method of playing their Lotto. Basically (I haven't paid that much attention), if you pay $2 for your ticket, you get an additional 10 million dollars. Three dollars is something like 25 million dollars.

Now, if we all agree that even the bare-bones 3 million dollar jackpot is a life-changing amount of money, why would anyone spend three dollars to get a 25 million dollar bonus - but only if their one ticket matches - rather than buy three separate tickets, which triples their chances of winning the "measly" 3 million in the first place?

Its the same system of decision making that leads people to shell out for tickets when the jackpot reaches astronomical proportions. They think they are going to win or they decide that the chance is worth it only for a gigantic payoff.

Its counter-logical decision making to pay $3 for a lottery ticket instead of buying three separate tickets. But I'm sure people are doing it.

If the odds somehow were 50/50 for one incident, then it would be 50/50 for 1000 incidents too, since they only consist of 1000 single incidents counted together. I can't understand how you can think that it's a 50% chance for one single incident, but a 67 % chance for 100 or 1000 of them. That's just illogical.

This is where it all breaks down. I too can't understand how Jim keeps stating that he understands that the odds improve by switching yet continues to think that in a singular event the odds are somehow 50-50. If they are 67-33 over 1000 iterations, then they are 67-33 for every single iteration.

Dear, sweet Dan -

I am not saying that the odds of switching having a successful outcome are ever 50-50 instead of 67-33.

But let's look at this non-linearly and/or non 3-D: What are the odds that the 67-33 odds will come to pass in a one-time only scenario with only two possible outcomes? 67-33? No, that's the odds for the choice itself being successful, not the odds for the choice actually materializing in real time.

When switching, the odds of winning are 67-33 in a one-time only scenario with only two possible outcomes.

Your last statement has no logical meaning.

This is not about the odds, it's about the odds of the odds.

This makes no sense. The respective odds apply to every single time that you apply the two different strategies.

Can you guarantee that in a one shot game that the 67% outcome will be the one that materializes? No, of course not.

No one has ever disputed this. It is understood that switching carries no guarantee of winning. But everyone else seems to understand that switching improves your chances of winning, by doubling those chances.

[Aggie87]

Jim's 50/50 thing isn't related to any odds. He's just saying he has two choices - stay or change.

[JSngry]

There is the so-called law of large numbers in statistics that says that if you play the switching strategy enough times in a row, the the number of times that you win the car will converge to 2 out of 3. If you only play a few times, there is a strong probability that you won't get the car two out of three times. But that is something different entirely than assessing the odds of a single event.

Well ok, you're talking prose, I'm mumbling poetry, but yeah, the earlier expressed notion that you should be surprised if you "smart" pick doesn't pan out the one and only time you get to use it is kinda...laughable to me.

Like I said, I've lost big betting on hands with "favorable odds" and won big by sometimes chasing after things that go against the odds. "Beating the odds" is not always "luck", ya' know. Some cats really do have an intuitive sense about which way the wind's blowing at any point in time, and I've lost money to some of them. But then again, there's been times when I've had that intuition working too, and I've won money from people who didn't. and on the whole, hey...I ain't poor, let me put it that way, and if even though that's in no way a result of gambling, it's also not a result of not gambling, if you know what i mean.

Odds, in a non-"theoretical" sense, are really just averages, which means that they don't always pan out, which means that sometimes, sometimes, counter-logical play is going to defeat logical play. I know that bugs people who like to think that the universe is a benign, non-fluid oasis of stasis, orderly place and if you just wait it all out you'll come out a winner, but...it is what it is, and what it is ain't no Sure Thing, ever.