Three Little Boxes

[Kathryn]

Phrasing can affect how the maths need to be done to show the result. Specifically, the instructions for which the host is supposed to operate are sometimes assumed differently. I don't know all variations, but the ones I'm familiar with all lead to the same answer.

Can you provide an example of something that hypothetically increases odds without really increasing odds? I don't know what that means.


If you're interested in the reality of winning, then it would be in your favor to switch. Mathematics are useless if they don't have applications- this one is applicable.

The math itself is rather straightforward once applied, but people still often don't believe it (it's rather unintuitive at first glance), so simulations can be run to show how you're better off switching.

Surely there have been some "real life" studies that have shown that mathematically speaking, switching actually DOES increase the number of wins.

 

[Penumbra]

Surely there have been some "real life" studies that have shown that mathematically speaking, switching actually DOES increase the number of wins.

Have you taken out a deck of cards to try it?

 

[Revoltingest]

Surely there have been some "real life" studies that have shown that mathematically speaking, switching actually DOES increase the number of wins.

Just my luck....when I tried to demonstrate the "correct" result to a friend, he ended up proving the wrong result.
With enuf trials, it would give a better result, but he decided to quite while he was ahead. Lousy probability!

 

[Penumbra]

Just my luck....when I tried to demonstrate the "correct" result to a friend, he ended up proving the wrong result.
With enuf trials, it would give a better result, but he decided to quite while he was ahead. Lousy probability!

The easiest say to show it (other than through math), is to write a program that does it.

Doing it by hand is possible too, but it takes more time to do it with a sufficiently large sample size. (One would have to be quite curious to do it by hand with a large enough sample size.)

 

[Debater Slayer]

Switching the box would increase the probability of having the gold by 1/3, so statistically speaking, it would make more sense to take the green box instead.

However, as there is still the probability that the green box doesn't contain the gold - albeit a reduced probability - the statistics only serve to slightly tip the odds in your favor rather than ensure your having the 'right' box.

 

[Kathryn]

Have you taken out a deck of cards to try it?

No, I don't even own a deck of cards.

Are there any studies that have shown, not the statistical ODDS, but the actual WINS using this tactic?

 

[Badran]

This is the part that explained it the best to me in the Wiki article (that and the table):

Simply put, if the contestant picks the goat (to which two of the three doors lead) they will win a car by switching as the other goat can no longer be picked, while if the contestant picks the car (to which one door leads) they will not win the car by switching. So, if you switch, you win the car if you originally picked the goat and you won't if you picked the car, and as you have a 2 in 3 chance of originally picking a goat you have a 2 in 3 chance of winning by switching.

The Wiki article

Since you're more likely to pick wrong in the first place, when all options are eliminated from the other choices that you haven't picked, and there's only one left, its more likely that that would be the right choice rather than the one you originally picked.

Also as the article said, one can appreciate or see the issue more clearly when the number of choices given is increased.

 

[Kathryn]

The easiest say to show it (other than through math), is to write a program that does it.

Doing it by hand is possible too, but it takes more time to do it with a sufficiently large sample size. (One would have to be quite curious to do it by hand with a large enough sample size.)

Well, you'd think with all the brouhaha over the years, someone would have taken the time to actually prove it with results.

I mean, my gosh, people stand around with their hands on cars for five days trying to win the car. They kiss for days to break a world record.

C'mon, man!

 

[Penumbra]

Well, you'd think with all the brouhaha over the years, someone would have taken the time to actually prove it with results.

I mean, my gosh, people stand around with their hands on cars for five days trying to win the car. They kiss for days to break a world record.

C'mon, man!

They have proven the results. It's mathematically proven. And then simulations have been done- going through trials to show that it works in practice to settle any remaining skepticism of people.

 

[Kathryn]

They have proven the results. It's mathematically proven. And then simulations have been done- going through trials to show that it works in practice to settle any remaining skepticism of people.

So no "real life" studies then?

Sorry, but you're talking to a person who is not mathematically oriented. I'm a logical person but I have very limited patience with hypothetical scenarios. My question is always, "So what's the real life application?"

I freely admit that this limits my perspective. I also believe that everyone's perspective is limited by their own set of talents and weaknesses.

What impresses me is that so many intelligent, educated, rational people react so strongly to this riddle. I pay attention to that sort of reaction. It's interesting to me.

 

[Debater Slayer]

Well, you'd think with all the brouhaha over the years, someone would have taken the time to actually prove it with results.

I mean, my gosh, people stand around with their hands on cars for five days trying to win the car. They kiss for days to break a world record.

C'mon, man!

I believe someone in this thread has already shown the mathematical proof for it, but here is another one that also illustrates the point:

Let X be the brown box, Y be the red box, and Z be the green box.

P(X) + P(Y) + P(Z) = 1

P(X) = 1 - P(Y) - P(Z)

P(X) = 1 - 1/3 - 1/3

P(X) = 1 - 2/3

P(X) = 1/3

This is the default assumption before opening any of the three boxes to check their content.

Now, plugging the result that the red box (Y) turned out to be containing sand into the above equation:

P(Z) = 1 - P(Y) - P(X)

P(Z) = 1 - 0 - 1/3

P(Z) = 1 - 1/3 = 2/3

Therefore, switching would make it more likely that the box you have chosen (the green box) is the one that contains the gold with an extra probability of 1/3 in its favor.

 

[Badran]

"So what's the real life application?"

The real life application is to try it enough times to see that inevitably switching results in more number of times of picking the right one than sticking to your original choice.

Put differently, why wouldn't you switch in the first place?

If you were given 10 choices instead of three, or a million, would you still stick to your original choice?

You are much more likely to pick wrong at first, so its the better choice to switch, since its the scenario that has a higher probability of you picking correctly.

 

[Father Heathen]

Surely I'm missing something here.

By switching, you only lose if you had picked the gold, which was a 1 in 3 chance. Thus winning by switching is a 2 in 3 chance.

 

[Kathryn]

By switching, you only lose if you had picked the gold, which was a 1 in 3 chance. Thus winning by switching is a 2 in 3 chance.

Sorry - I don't see how that is.

It seems to me that switching would give a 50/50 chance because now you know more than you did before. But not a 2/3 chance.

But you'll probably lose patience trying to explain math to me. I've done quite well in several fields which actually involve quite a bit of math (real estate and banking) without having the slightest bit of warmth toward the subject.

 

[dawny0826]

Sorry - I don't see how that is.

It seems to me that switching would give a 50/50 chance because now you know more than you did before. But not a 2/3 chance.

But you'll probably lose patience trying to explain math to me. I've done quite well in several fields which actually involve quite a bit of math (real estate and banking) without having the slightest bit of warmth toward the subject.

Kathryn, I'm with you.

It's either one or the other box at this point.

I'd buy my own freaking purple box of silver earrings that make jingling noises and call it a day.

 

[atanu]

By switching, you only lose if you had picked the gold, which was a 1 in 3 chance. Thus winning by switching is a 2 in 3 chance.

Funniest thing, however, is that that 1/3 chance would fructify when that has to.

 

[Circle_One]

This was done on Mythbusters, they showed that keeping the brown box ended in gold more times than switching to the green box.

 

[Kathryn]

The real life application is to try it enough times to see that inevitably switching results in more number of times of picking the right one than sticking to your original choice.

Put differently, why wouldn't you switch in the first place?

If you were given 10 choices instead of three, or a million, would you still stick to your original choice?

You are much more likely to pick wrong at first, so its the better choice to switch, since its the scenario that has a higher probability of you picking correctly.

I didn't say I wouldn't switch. I just don't see how it's advantageous either way - to switch or keep the same choice seems like a 50/50 chance to me.

 

[Kathryn]

Kathryn, I'm with you.

It's either one or the other box at this point.

I'd buy my own freaking purple box of silver earrings that make jingling noises and call it a day.

Right on, sister!

 

[Penumbra]

This was done on Mythbusters, they showed that keeping the brown box ended in gold more times than switching to the green box.

I looked it up, and the result was the exact opposite of that.

When presented with the Monty Hall Problem, people would be more likely to win if they changed their decision. Confirmed
They built a small-scale simulator to do 50 trials each, with Adam always switching his choice and Jamie never switching. Adam won far more often than Jamie did, and Jamie explained the reason: because the player has a 2/3 probability of choosing a losing door at first, switching turns the odds in his favor.

MythBusters Results — Outcomes from all MythBusters Episodes


Mythbusters went through the trials and it worked out as the math says it would.