Another logic riddle, a lot tougher, builds on principles from the others

[meaner gene]

You step up to a carnival booth. The carney running the game has 3 numbered cups upside-down on the table in front of you. He tells you there is $300 under one of them, and nothing under the other two. He knows what is under each cup.

For $100, you get to choose a cup to be flipped and keep whatever is underneath.. You pay your $100 and point to Cup #1.

The carney flips over Cup #3 to reveal nothing is under the cup. He then asks you if you would like to keep your choice or change it to Cup #2.

What do you do? Does it even matter whether or not you switch your choice from Cup #1 to Cup #2?

This was done on Mythbusters. It was a recreation of the classic "lets make a deal", where a person was offered, Curtain number one, curtain number two or the box.

And after choosing one of them, Monty Hall revealed one of the unchosen to be "empty", and gives the chance for the person to switch or stick with their original choice.

The fact is the outcome is advantaged by switching each time. In short, the original pick had odds of 1 in 3 of being correct. But by choosing from the two remaining choices, the odds go up to 1 in 2.

This is because no matter if you chose the winning cup or not, the person running the game can ALWAYS turn over an empty cup. You can think of this by increasing the number of cups from 3 to some large number like 1,000.

Your odds of picking the right cup are 1,000 to 1. And the one running the game can turn over 998 empty cups whether you picked the right cup or not. But after flipping over 998 cups, the two remaining are 50/50. And by switching, you go from the original 1,000 to 1, to 50/50.

 

[Fort Fun Indiana]

This was done on Mythbusters. It was a recreation of the classic "lets make a deal", where a person was offered, Curtain number one, curtain number two or the box.

And after choosing one of them, Monty Hall revealed one of the unchosen to be "empty", and gives the chance for the person to switch or stick with their original choice.

The fact is the outcome is advantaged by switching each time. In short, the original pick had odds of 1 in 3 of being correct. But by choosing from the two remaining choices, the odds go up to 1 in 2.

This is because no matter if you chose the winning cup or not, the person running the game can ALWAYS turn over an empty cup. You can think of this by increasing the number of cups from 3 to some large number like 1,000.

Your odds of picking the right cup are 1,000 to 1. And the one running the game can turn over 998 empty cups whether you picked the right cup or not. But after flipping over 998 cups, the two remaining are 50/50. And by switching, you go from the original 1,000 to 1, to 50/50.

That explanation is not correct. If the odds of the "switch-to" second cup are 50/50, then there is no advantage at all to switching, as the odds of staying with your original choice are therefore also 50/50. So, it would not matter if you switched or not.

 

[meaner gene]

Noted.

Now, think about what you are stating:

You believe your odds have improved, because the carney showed you an empty cup after you made your bet. You believe Cup #1 had 1/3 odds, but it now has 1/2 odds. Because the carney showed you an empty cup.

Does that feel right to you?

Your original odds stay the same, as long as the carney can turn over an empty cup. Which in this scenario is always guaranteed.

 

[Fort Fun Indiana]

Your original odds stay the same, as long as the carney can turn over an empty cup. Which in this scenario is always guaranteed.

Maybe. But that is not what you described. If you have two cups remaining, and one cup has 50/50 odds, so does the other cup. There would be no advantage to switching.

 

[Anomalism]

Maybe. But that is not what you described. If you have two cups remaining, and one cup has 50/50 odds, so does the other cup. There would be no advantage to switching.

It's not 50-50. It's 1/3-2/3.

 

[meaner gene]

That explanation is not correct. If the odds of the "switch-to" second cup are 50/50, then there is no advantage at all to switching, as the odds of staying with your original choice are therefore also 50/50. So, it would not matter if you switched or not.

As I exampled. This is where expanding the number of cups makes the advantage of switching clearer.

If you start out with 10 cups, your odds started (and remain) at 1 in 10.
He can always flip 8 empty cups, thus your odds don't change by him doing so.
But with only 2 cups left, the odds are 50/50 by choosing anew, but remain 1 in 10 by sticking with your original choice.

 

[Fort Fun Indiana]

But with only 2 cups left, the odds are 50/50 by choosing anew

No.

 

[Anomalism]

the odds are 50/50 by choosing anew

 

[Anomalism]

Noted.

Now, think about what you are stating:

You believe your odds have improved, because the carney showed you an empty cup after you made your bet. You believe Cup #1 had 1/3 odds, but it now has 1/2 odds. Because the carney showed you an empty cup.

Does that feel right to you?

Declare me the winner already.

 

[Fort Fun Indiana]

Declare me the winner already.

But you haven't shown any work, whether or not you gave the right answer.

 

[Anomalism]

But you haven't shown any work, even if you gave the right answer. Still just collecting answers, I guess.

Cup 1 doesn't change odds just because you lifted cup 3. It's still 1/3. However if you switch to cup 2 your odds become 2/3, because one of the three cups is gone now, which means you're gambling on a 2/3 with cup 2 vs the original 1/3 with cup 1. It's still the same game you were playing with 3 cups even if one of them has been removed.

Statistically it's a better choice to pick cup 2.

 

[Fort Fun Indiana]

Cup 1 doesn't change odds just because you lifted cup 3. It's still 1/3. However if you switch to cup 2 the odds become 2/3, because one of the three cups is gone and you're now gambling on a 2/3 vs the original 1/3.

You say one of the cups is gone (we know), and that the answer is to switch, because your odds after switching are 2/3.

Noted.

You sure? Why?

 

[Anomalism]

You sure?

Yes.

Why?

It's still the same three cup game even if you removed one. Picking from your original position of 1/3 is inferior to picking from a new position of 2/3. You've removed one of the three cups and are also now choosing a second one.

 

[Fort Fun Indiana]

Cup 1 doesn't change odds just because you lifted cup 3. It's still 1/3. However if you switch to cup 2 your odds become 2/3, because one of the three cups is gone now, which means you're gambling on a 2/3 vs the original 1/3. It's still the same game you were playing with 3 cups even if one of them has been removed.

Statistically it's a better choice to pick cup 2.

Better after adding to it.

Your answer is recorded.

 

[meaner gene]

You say one of the cups is gone (we know), and that the answer is to switch, because your odds after switching are 2/3.

Noted.

You sure? Why?

Maybe this is a clearer explanation.

Three cups. You chose one of them, your odds are 1 in 3.
The alternate is if you switch, and choose the other 2 cups (hear me out) collectively
those two cups have odds of 2 in 3.

But you know that from those 2 cups, the carney will turn over an empty one, so that the one remaining cup inherits the same 2/3 odds.

 

[Fort Fun Indiana]

Maybe this is a clearer explanation.

Three cups. You chose one of them, your odds are 1 in 3.
The alternate is if you switch, and choose the other 2 cups (hear me out) collectively
those two cups have odds of 2 in 3.

But you know that from those 2 cups, the carney will turn over an empty one, so that the one remaining cup inherits the same 2/3 odds.

Gotcha.

 

[meaner gene]

Gotcha.

Is that gotcha as in "got you" / understand
Or gotcha as in gotcha! (you're wrong)

 

[Fort Fun Indiana]

Is that gotcha as in "got you" / understand
Or gotcha as in gotcha! (you're wrong)

I understand your explanation.

 

[Grumblenuts]

At the start the odds are 1/3. After one cup is revealed empty the odds of either remaining cup containing the $300 increase to 1/2.

 

[Anomalism]

Better after adding to it.

Your answer is recorded.

I'm sleepy. haha I tried my best. A lot of people probably won't understand even after having it explained to them.