ok then, since we've all got our brains warmed up on stu's thread, lets see what you guys make of this one...

 

a contestant on a gameshow is asked to pick one of three mystery doors. behind one of the doors is a prize (don't know what but i'm sure its dead good!)

 

after the contestant picks one of the doors, the host then opens one of the doors that wasn't picked to reveal that there is nothing behind it. the contestant is then offered the choice to stay with his/her original choice or switch to the other one.

 

does the contestant have a better, worse or equal chance of winning if he/she chooses the other box? discuss.

Better

Depends...

On paper, when the contestant first selected the door he had a 33.3% chance of winning the price. Now that he has removed one of the choices, he had a 50% chance of winning.

Realistically....

He is asked if he wants to swap after one of the doors have been eliminated, so it makes no odd's, whether the swaps or stays the same. He still only has 2 doors to choose from so the chances of sinning at this stage stay the same (i.e. 1 in 2)

However if you mean does he have a better chance of winning from the start (i.e. 1 in 3).Then yes it will be better, as the chances of swapping to 1 other door which will either be nothing or a prize (i.e. 2 choices) is a better chance than 2 other doors which will be either nothing and nothing or nothing and prize.

:confused:

 

It's better to swap because the chance of choosing the correct door first go is 1/3, so the chances of getting it wrong are 2/3.

 

So, if you are allowed to swap, and you're told which of the other 2 is empty, then sticking with what you have still gives you a 1/3 chance of being right, but changing gives you the 2/3 chance of being right.

Originally posted by Dave Marley

:confused:

 

It's better to swap because the chance of choosing the correct door first go is 1/3, so the chances of getting it wrong are 2/3.

 

So, if you are allowed to swap, and you're told which of the other 2 is empty, then sticking with what you have still gives you a 1/3 chance of being right, but changing gives you the 2/3 chance of being right.

 

Given that there are NOW 2 possible boxes to choose from, it is 1 in 2, whether you change or not.

 

It's a new situation as soon as an empty box is revealed. 1 in 2 chance, from 1 in 3 previously.

 

What if previously - unbeknown to us - a fourth option had been available then one empty box taken away, does that make it all of a sudden 1 in 4? No. ;)

 

Pete

Originally posted by pete shrimp

Given that there are NOW 2 possible boxes to choose from, it is 1 in 2, whether you change or not.

The question is whether you got it right first time or not. You have extra information which needs to be taken into account, so it's not just a straight 50/50.

Yea but it all depends who the game host is.

 

If It was Duffer then all the boxes would be empty;)

Originally posted by Dave Marley

The question is whether you got it right first time or not. You have extra information which needs to be taken into account, so it's not just a straight 50/50.

 

Aye but given that you don't know whether you've chosen the right one, of course its 50/50.

 

Its 2 separate scenarios:

 

a) 3 boxes of which one is a winner -> 1/3

then take away a wrong box,

b) 2 boxes of which one is a winner -> 1/2

 

It doesn't matter that you can change your mind, you could do that anyway in your own head. Does that fact that you speak your mind change the odds? No.

Originally posted by pete shrimp

Aye but given that you don't know whether you've chosen the right one, of course its 50/50.

No it isn't!

 

Take another example. There are 1000 boxes and you get to choose 1. Your chances of getting it right are only 1/1000.

 

The host opens up 998 empty boxes, leaving you with the one you originally chose and one other. Do you really think that it's just as good to stick with the original box you chose as it's a 50/50 chance?

 

By the host opening 998 empty boxes, you're being given the opportunity to take the original 1/1000 chance, or accept that you were almost certainly wrong in the first place and take the 999/1000 chance!

Originally posted by Dave Marley

No it isn't!

Yes it is :D

 

Take another example. There are 1000 boxes and you get to choose 1. Your chances of getting it right are only 1/1000.

Yep, ok.

 

The host opens up 998 empty boxes, leaving you with the one you originally chose and one other. Do you really think that it's just as good to stick with the original box you chose as it's a 50/50 chance?

Ok, you've now got 2 boxes left, either is a 50/50 chance! It doesn't matter how many other boxes there were to start with, the quizmaster was goona do away with them anyway, leaving 2!

 

By the host opening 998 empty boxes, you're being given the opportunity to take the original 1/1000 chance, or accept that you were almost certainly wrong in the first place and take the 999/1000 chance!

Actually this sentance seems pretty rigid, damn I'm confused now :D.

 

Hmmm maybe some humble pie...:D

Originally posted by pete shrimp

Hmmm maybe some humble pie...:D

I didn't catch that. It was very small! :D

The host opens up 998 empty boxes, leaving you with the one you originally chose and one other. Do you really think that it's just as good to stick with the original box you chose as it's a 50/50 chance?

 

Yes!

 

By the host opening 998 empty boxes, you're being given the opportunity to take the original 1/1000 chance, or accept that you were almost certainly wrong in the first place and take the 999/1000 chance!

 

Pants!

Originally posted by Peter Hood

Yes!

 

 

 

Pants!

Did you say 'Pants' because you just realised you were wrong, or are you still arguing the toss? :D

LOL You wouldn't believe I've a degree in Maths would ya? :D

 

Suppose it just needed illustrating well. Eg in the 1000 boxes scenario, if the producers know the prize is in say box 337, and you choose box 870.

 

They take away all the others, leaving 337 & 870...

 

Best to swap I think...:D

 

Ok what if there were 2 boxes and the quizmaster took away the right one...

Originally posted by pete shrimp

Ok what if there were 2 boxes and the quizmaster took away the right one...

I'd sue! :D

LOL Why am i going on about boxes? I'm sure it started off as doors :confused: :D

 

Is Des O'Connor the quizmaster?

Still arguing the toss! You were originally give a 1/1000 chance of being right. Once the boxes are opened you are then playing a new game and being given a 1/2 chance. The 998 opened boxes are irrellevant.

Originally posted by Peter Hood

Still arguing the toss! You were originally give a 1/1000 chance of being right. Once the boxes are opened you are then playing a new game and being given a 1/2 chance. The 998 opened boxes are irrellevant.

No they aren't! ;)

 

Think of it this way. You have 1000 boxes/doors and choose 1.

 

The host gives you the chance of opening the one you originally chose, or of opening all the others. Which would you choose?

 

It's exactly the same principle, but you just have to open the extra 998 doors/boxes yourself instead of him doing it before asking you. :cool:

Originally posted by Peter Hood

Still arguing the toss! You were originally give a 1/1000 chance of being right. Once the boxes are opened you are then playing a new game and being given a 1/2 chance. The 998 opened boxes are irrellevant.

 

Nah cos the one you picked first is probably wrong, so the other one of the remaining 2 is probably right, so change your mind.

 

1 - probably wrong = probably right ;)

congratulations dave (and pete in the end) - it is better to swap. its already been explained pretty well but in a nutshell:

 

three boxes, A B C

 

contestant picks box A, it has a 1/3 chance of winning

 

box B is then shown to be empty but this does not change te odds of the original bet

 

the choice then becomes

 

1/3 that the prize is in box A

or

2/3 that the prize is not in box A

 

the only other option is box C so the odds of it winning are 2/3

[child]

 

See Dave I told you!! :D

 

[/child]

Nope! If the host offers you the chance to either open 999 boxes or 1 box he is offering you 999/1000 or 1/1000 respectively so obviously you take the 999/1000 odds.

 

If the host opens the 998 boxes he is giving himself a 998/1000 chance of finding the correct box, you have a 1/1000 chance.

 

If the host opens 998 empty boxes and then offers you the chance to pick either of the remaining boxes he is offering you new odds of 1/2.

 

Pete,

The first box you picked is probably wrong , when there was 1000 unopened boxes there was a 1/1000 chance of it being correct but the other unopened box also only had a 1/1000 chance of beind correct at that time. However now that there are only two boxes left they both have a 1/2 chance of being correct.

 

As an aside if you toss a coin 10 times and get a head every time whats the odds of an head on the 11th toss.

Originally posted by Peter Hood

Nope! If the host offers you the chance to either open 999 boxes or 1 box he is offering you 999/1000 or 1/1000 respectively so obviously you take the 999/1000 odds.

 

If the host opens the 998 boxes he is giving himself a 998/1000 chance of finding the correct box, you have a 1/1000 chance.

The host is removing only empty boxes. There is no chance of him opening a full box as he knows which one is full and doesn't open that one!

 

Pete,

The first box you picked is probably wrong , when there was 1000 unopened boxes there was a 1/1000 chance of it being correct but the other unopened box also only had a 1/1000 chance of beind correct at that time. However now that there are only two boxes left they both have a 1/2 chance of being correct.

Aaaarrrghh! :eek:

 

As an aside if you toss a coin 10 times and get a head every time whats the odds of an head on the 11th toss.

That's a different question! :)

[

box B is then shown to be empty but this does not change te odds of the original bet

 

Pants!

 

Three boxes A,B and C. Contestant picks A, host opens B and C, both empty (but this does not change the odds of the original bet!) ie. 1/3 chance that A is correct. Methinks not, any of you guys up for a card school!

Originally posted by Peter Hood

Methinks not, any of you guys up for a card school!

I'll be there in an hour. Bring plenty of cash! :D

Originally posted by ste

congratulations dave (and pete in the end) - it is better to swap. its already been explained pretty well but in a nutshell:

 

three boxes, A B C

 

contestant picks box A, it has a 1/3 chance of winning

 

box B is then shown to be empty but this does not change te odds of the original bet

 

the choice then becomes

 

1/3 that the prize is in box A

or

2/3 that the prize is not in box A

 

the only other option is box C so the odds of it winning are 2/3

 

PANTS

 

That's complete pants !!!

So you are saying that if he swaps to Box C he has more chance of winning. That's complete Kak mate.

Whether he changes his mind or not of course he has a better chance of winning if you take one of the boxes away.

But you said "does the contestant have a better, worse or equal chance of winning if he/she chooses the other box?" Meaning Box C.

He knows that Box B is empty so he has a choice of box A or C. Why would changing from box A to C increase his chances of winning ?