Another maths problem! Three doors
Discussion
Following on from the other maths problem thread I thought I'd try this one (bit of a classic):
You get through to the final round of a gameshow. You have a choice of three doors; behind one is the big prize, behind the other two nothing. You choose a door at random. The host then opens one of the other two doors to reveal nothing. The host then gives you the opportunity to stick with the door you originally chose, or to switch to the other, unopened door.
The question is, in order to get the best chance of winning the big prize should you switch, or would it not make any difference?
You get through to the final round of a gameshow. You have a choice of three doors; behind one is the big prize, behind the other two nothing. You choose a door at random. The host then opens one of the other two doors to reveal nothing. The host then gives you the opportunity to stick with the door you originally chose, or to switch to the other, unopened door.
The question is, in order to get the best chance of winning the big prize should you switch, or would it not make any difference?
Mathematically, you should switch. But that doesn't mean you win.
Initially, 3 doors. So each door has 1/3 chance of being the correct one.
Once you pick a door, there is a 2/3 chance that the prize is behind one of the other doors (that you didn't select).
Given the host open one of the doors you didn't select, revealing nothing, that means there is now 2/3 chance of the prize being behind the other door that you didn't pick, rather than the 1/3 that door initially had.
Initially, 3 doors. So each door has 1/3 chance of being the correct one.
Once you pick a door, there is a 2/3 chance that the prize is behind one of the other doors (that you didn't select).
Given the host open one of the doors you didn't select, revealing nothing, that means there is now 2/3 chance of the prize being behind the other door that you didn't pick, rather than the 1/3 that door initially had.
Edited by JatHanspal on Thursday 15th November 15:47
KrazyIvan said:
Makes no difference, you had 1 in 3 chance.
You now have a 1 in 2 chance.
Original you were 33% likely to have picked the right door, now you are 50% likely to have picked the right door.
Think about what you've said, and apply it to 1000 doors. You pick door 117. The host opens all the other doors apart from 883. Do you switch. Of course you do. At the outset you have a 1 in1000 chance of 117 being the right door. That hasn't changed. It's not now 50/50. You now have a 1 in 2 chance.
Original you were 33% likely to have picked the right door, now you are 50% likely to have picked the right door.
It's 1 in a 1000 117 is right, and 999 in a 1000 883 is right.
It's the Monty Hall problem. You should switch. Still a 2 in 3 chance the door you didn't pick is the right one.
What the OP didn't say, which is important, is that the host knows where the prize is.
Edited by TwigtheWonderkid on Thursday 15th November 15:53
Edited by TwigtheWonderkid on Thursday 15th November 15:55
Nanook said:
Not quite.
If you don't change your mind, you still only have a 1 in 3 chance.
If you do change your mind, you have a 2 in 3 chance though.
Correct. The reason the odds change is that you get new information. The host knows where the prize is so he can always open an empty door.If you don't change your mind, you still only have a 1 in 3 chance.
If you do change your mind, you have a 2 in 3 chance though.
If the host tells you that one of the doors you have picked is empty but not which one, then you would still have a 1 in 3 chance, however once he opens and shows you and ampty door, and then gives you a chance to switch you now have a 1 in 2 chance as you cannot switch back to the known empty door.
In other words when you are asked if you want to switch it, is the same as being asked to choose a door from the start but now you only have 2 doors to choose from.
So as I said you have 1 in 3 on the 1st choice and 1 in 2 on the 2nd choice.
All assuming you cannot switch to the known door.....unless it's on channel 5
In other words when you are asked if you want to switch it, is the same as being asked to choose a door from the start but now you only have 2 doors to choose from.
So as I said you have 1 in 3 on the 1st choice and 1 in 2 on the 2nd choice.
All assuming you cannot switch to the known door.....unless it's on channel 5
KrazyIvan said:
So as I said you have 1 in 3 on the 1st choice and 1 in 2 on the 2nd choice.
When you first had the choice it was a 1 in 3 chance. When he removes a door from the pool, your door stays at 1 in 3, as you have no new information about your door.
The other door increases from 1 in 3 to 2 in 3 chance. (the chances must add up to 100% right?!)
ReallyReallyGood said:
When you first had the choice it was a 1 in 3 chance.
When he removes a door from the pool, your door stays at 1 in 3, as you have no new information about your door.
The other door increases from 1 in 3 to 2 in 3 chance.
THIS!!!When he removes a door from the pool, your door stays at 1 in 3, as you have no new information about your door.
The other door increases from 1 in 3 to 2 in 3 chance.
As I said. imagine 1000 doors. You pick 117, he opens all others apart from 883. Switch...of course you bloody switch. It's bound to be in 883, unless you got very lucky with your 117 guess.
TwigtheWonderkid said:
What the OP didn't say, which is important, is that the host knows where the prize is.
Does that matter though? If the host doesn't know where the prize is and picks one of the other two at random and it is revealed to be empty before you choose whether to switch, then how does that differ from the situation where the host does know where the prize is and reveals an empty door? Roger Irrelevant said:
TwigtheWonderkid said:
What the OP didn't say, which is important, is that the host knows where the prize is.
Does that matter though? If the host doesn't know where the prize is and picks one of the other two at random and it is revealed to be empty before you choose whether to switch, then how does that differ from the situation where the host does know where the prize is and reveals an empty door? TwigtheWonderkid said:
deckster said:
Lotus Notes said:
It's a Monty Hall problem called goat or Ferrari.
Always take the offer.
And what if you'd rather have the goat than the Ferrari?Always take the offer.
Now, anybody want a goat?
TwigtheWonderkid said:
Roger Irrelevant said:
TwigtheWonderkid said:
What the OP didn't say, which is important, is that the host knows where the prize is.
Does that matter though? If the host doesn't know where the prize is and picks one of the other two at random and it is revealed to be empty before you choose whether to switch, then how does that differ from the situation where the host does know where the prize is and reveals an empty door? Nanook said:
Roger Irrelevant said:
I've just popped back to say that you were right all along Twig - I thought about this on the way home last night and it does matter whether the host knows where the prize is, or if he chooses at random. If the host does know where the prize is then by switching your chance of winning is 2/3. If the host doesn't know and just happens to reveal an empty door, it's 50:50 between the door you originally chose and the one that's left. I then got on to thinking what the case would be if you didn't know whether the host knew or not, but fortunately I got home before my brain melted!
What?When the host opens that door to reveal nothing, and you decide to swap to the other remaining door, you think the chances of you now having the correct door can vary between 1:1 or 1:3, depending solely on whether or not the host knows what's behind each door?
You'll have to explain that.
In probabalistic terms the situation is exactly equivalent to if the host opened a door at random to reveal it was empty before you chose your door, and in that case it's easy to see that it's 50:50.
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