If it's a matter of exercising, here comes my question

You are a contestant on the MegaPrize game show. The host shows you three boxes, labelled A, B, and C. Inside one of the boxes is a cheque for a large sum of money. All you have to do to win is pick the right box. The host asks you to choose one of the boxes, and let's say you pick box A. If this were all you had to do then you would obviously have a one-in-three chance of winning.
But there is a twist. The host now opens one of the other boxes, say box C, and shows you that it is empty. He now invites you to change your mind, and pick box B instead of box A. Should you stick with box A or change to box B? Does changing your choice affect your chance of winning, or doesn't it matter?

Please think carefully, it's not an easy question; a lot of Ph.D students gave the wrong answer.

If it's a matter of exercising, here comes my question

You are a contestant on the MegaPrize game show. The host shows you three boxes, labelled A, B, and C. Inside one of the boxes is a cheque for a large sum of money. All you have to do to win is pick the right box. The host asks you to choose one of the boxes, and let's say you pick box A. If this were all you had to do then you would obviously have a one-in-three chance of winning.
But there is a twist. The host now opens one of the other boxes, say box C, and shows you that it is empty. He now invites you to change your mind, and pick box B instead of box A. Should you stick with box A or change to box B? Does changing your choice affect your chance of winning, or doesn't it matter?

Please think carefully, it's not an easy question; a lot of Ph.D students gave the wrong answer.

What is this..Quiz Friday?!

I would stick to option A because I'm a secure person and generally speaking, the first answer is always the right one....and I had a good feeling about A.

You'd change - seen it in a film somewhere - you increase your chances or getting right by changing for some illogical (or rather logical) reason that I can't explain...

You'd change - seen it in a film somewhere - you increase your chances or getting right by changing for some illogical (or rather logical) reason that I can't explain...

The reasons are logical, and if you actually perform an experiment along these lines, you get exactly those results. But it is counter intuitive and I still don't understand the reasoning.

The host now opens one of the other boxes, say box C, and shows you that it is empty.

I think the key is in the wording, the host deliberately picks an empty box.

for those having trouble understanding, let's imagine instead of 3 boxes, there are 100 boxes. you pick one. the host then opens 98 empty boxes and invites you to swap...

The Quantum scientist would argue that the cheque does not exist until the box is opened... Therefore prior to opening, you are playing a game to win something that doesn't exist.

Now that I think of it, none of you are real. You exist only in my mind. If you think you exist, it's because in my mind I think that you think you exist.

The reasons are logical, and if you actually perform an experiment along these lines, you get exactly those results. But it is counter intuitive and I still don't understand the reasoning.

Oh, I know it is logical - but it is so logical that I don't get it, so from my point of view it is illogical. Get it? 'cos I still don't...

I think the key is in the wording, the host deliberately picks an empty box.

for those having trouble understanding, let's imagine instead of 3 boxes, there are 100 boxes. you pick one. the host then opens 98 empty boxes and invites you to swap...

You choose 1 box: 33% chance that you got it right, versus 66% chance that you got it wrong.

Host opens one (empty) box: that means that there is now a 66% chance that the remaining box that you didn't choose has the prize, while your choice of box remains at 33% chance of getting it right.

I would stick to option A because I'm a secure person and generally speaking, the first answer is always the right one....and I had a good feeling about A.

The book I was talking about above has a mathematical explanation which I don't understand...

But it also has this little diagram which explains it. I'll attach the page so you can see both.

As you can see, if you change, you have 2 changes of getting a car, but only one of getting a goat. So you should change.

Statistically, swapping is the logical answer as explained. If you were correct the first time, then staying with that selection would be the correct option i.e. Do you trust your instinct?

If it's a matter of exercising, here comes my question

You are a contestant on the MegaPrize game show. The host shows you three boxes, labelled A, B, and C. Inside one of the boxes is a cheque for a large sum of money. All you have to do to win is pick the right box. The host asks you to choose one of the boxes, and let's say you pick box A. If this were all you had to do then you would obviously have a one-in-three chance of winning.
But there is a twist. The host now opens one of the other boxes, say box C, and shows you that it is empty. He now invites you to change your mind, and pick box B instead of box A. Should you stick with box A or change to box B? Does changing your choice affect your chance of winning, or doesn't it matter?

Please think carefully, it's not an easy question; a lot of Ph.D students gave the wrong answer.

Initially it's a 1:3 chance of winning. Then with choice C being eliminated, the odds have gone up to 1:2. The maneuver of eliminating C makes you second guess your choice, and therefore the odds of choosing incorrectly by picking B goes up.

The book I was talking about above has a mathematical explanation which I don't understand...

But it also has this little diagram which explains it. I'll attach the page so you can see both.

As you can see, if you change, you have 2 changes of getting a car, but only one of getting a goat. So you should change.

The model assumes that changing increases a chance of getting a car, but not of getting a goat. That's not logical, because changing could still result in getting a goat.