View Full Version : Puzzle
This is an old puzzle, but I still like it:
A TV quiz programme involves three closed boxes. In one of these boxes is a huge diamond ring worth a million pounds; the other two boxes are empty. The contestant doesn't know which box the ring is in, but the quizmaster (let's call him Bill) does know.
The contestant chooses a box, and Bill immediately responds by opening one of the other boxes -- one that he knows is empty. If the contestant has chosen the correct box, Bill opens one of the two empty boxes at random.
After Bill has shown that the box he opens is empty, the contestant is given a chance to change her choice of box.
Should she change?
Or should she stay with her first choice?
Does it make any difference? If so, why?
What is the likelihood of winning the ring if she sticks with her first choice? What if she decides to change?
If you already know the answer, please don't post it here: give everyone else a chance to solve it. Similarly, please don't post solutions from from the web, or links to any websites that might contain the answer. I got it wrong at first. It's easy if you use a bit of lateral thinking.
champagnebaby
14-Apr-05, 22:27
I'm stumped :roll:
I give up - off to google it
champagnebaby
14-Apr-05, 22:32
Got it!
Tut, why didn't i think like that :roll:
I'm stimped - any clues? I don't want to give in and google.
Surely too easy :confused
Think I got it! as you say lateral thinking every time.
gravedigga
16-Apr-05, 02:49
Yup! lateral thinkin is the key!!
I started thinkin o it as a trick question . . . . . . . . . . then i gave up and i thot o it in simple terms n got it right - resaon i know i got it right is cos i googled it after a whilie
I'm know I'm not a lateral thinking type person so forgive me, but surely it's just the odds of winning that change? You had a one in three chance and now you have a 50/50 chance? :confused
When will you post the answer DrSzin?
That's what i thought to begin with fifi, but was told it was wrong. Then I misread it, looking at it as a trick question and again came up with a wrong answer. Now I know the right answer, but (as I do) I would argue with it. I conducted my own research with three matchboxes,several times to test the theory, and I'm back to my origonal thoughts that it is 50/50, whatever the experts say.
Mr P Cannop
16-Apr-05, 13:59
phone a friend
I will post the answer eventually. But people seem to be having fun (not sure that's the right word!) trying to solve it, so I won't post the solution just yet.
In the meantime, I think it's easier to get the answer if you think about the chances of not winning rather than those of winning. If I say anything else I might give the game away.
BTW 50/50 is definitely not the right answer, and it's not a trick question at all. I could perhaps have worded the "at random" bit a little more clearly.
I too did a practical test to check my answer. If I remember rightly, I found that you need to do at least 30, perhaps more, tests to get an answer that is stable. Doing it (say) 10 times is not enough -- it's a very good way of learning that you can't reliably apply probability and statistics to just a few trials -- you need lots of data to do that.
sassylass
16-Apr-05, 17:25
Ok please help me figure by answering some questions:
Are the boxes transparent?
Is the diamond so huge as to protrude from the box?
Is the contestant blind?
Is the contestant allowed to shake the boxes?
How quickly does Bill respond and open the other box?
What words does one type into google to find the answer? :p
Thank you!
This is my own reckonning - However the chances change, the fact is, you are left with one empty box and one with the ring. You start with two chances in three of being wrong. That is reduced to one chance in two. now your chances are better, but it's still 50/50 isn't it?
It ain't 50/50 because based on the info you have been given you should know which box the ring is in ;)
So i'm given a choice of three boxes, I chose one, 1/3. I'm shown an empty one which reduces the odds to 1/2. And I should know which one it's in for sure?
You are given more info than that - you are told that the quizmaster Bill Knows which box the ring is in ;)
Naefearjustbeer
17-Apr-05, 13:48
easy I think,
What differnce does that make? I know he knows which one of the boxes contain the ring, and he has delibrately shown me an empty box, but all he has done is to reduce the odds from 1/3 to 1/2. the choice is still down to me. Unless he somehow gives the game away, but I suspect the solution is more scientific than that.
:roll: Reverse the logic Katrina and think of the options available to Bill when you choose your Box.
:confused But then again I could be wrong ;) :lol:
:roll: Reverse the logic Katrina and think of the options available to Bill when you choose your Box.
If you chose the right box, the options open to Bill is either one of the two empty boxes. If you chose an empty box, then Bill has only one option - the other empty box. But you don't know that - if you did then it's easy.
Lets try it this way. Pretend I am Bill. I will number the boxes 1 2 3. Now I know which box the ring is in. You chose a box and let me know.
~~Tides~~
17-Apr-05, 17:29
The contestant chooses a box, and Bill immediately responds by opening one of the other boxes -- one that he knows is empty. If the contestant has chosen the correct box, Bill opens one of the two empty boxes at random.
Am I right in thinking this is the imortant bit?
Naefearjustbeer
17-Apr-05, 23:14
ok its not so easy :confused
sassylass
18-Apr-05, 02:07
well I was thinking way too hard on this one.....it came to me in a flash, when I stopped pondering and relaxed my brain...hehehe good one Szin
I'm still arguing it's 50/50. When you start its 25/75 in Bill's favour. when bill removes one of the boxes your odds are doubled - which makes it 50/50.
Doesn't it?
:confused How many boxes are you dealing with Katrina?
:confused How many boxes are you dealing with Katrina?
three. 1,2 and 3 just like bill.
Check the sums on yer odds - not that it will effect the answer.
ok. 1 chance in 3. = 25/75. no? you are shown an empty box. Now there are two boxes left. one with the ring the other empty. 1 chance in 2. 50/50. no?
go ahaed, chose a box. 1,2 or 3 and we'll try it out.
Mind you, numeracy always was my weakest point......
After reading it a million times and reading your clue I think I now have it. The problem I had was I was looking for a way to find out 100% of the time but now I realise that's not possible. I understand now why it is not 50/50.
Well if it's not 50/50 it's pretty near when there's only three boxes to begin with. I wouldn't bet any money on being able to tell the right one. How about you scotsboy? Wanna put your money where your mouth is?
No problem, but it would need to be face to face and you woiuld have to follow verbatim the instructions as originally provided ;)
why would you need to be face to face? This implies that some body language gives the game away.
Let me go through this again. Just say you picked number three. I open number 2 cos i know that one is empty. Now you know it is either 1 or 3. I knows which one it is - you don't. but you say you do -tell me how.
ironic_clans
18-Apr-05, 20:38
is the answer 42 ?
Right. I think I've got your reasoning. Still pretty dodgy territory when ther's a million pounds at stake!
Ok, I think everyone has either solved this puzzle, or has tired of it. Here is a solution:
There are 3 boxes, therefore the contestant's initial guess will be right 1/3 of the time, and wrong 2/3 of the time.
If she always sticks with her initial choice she will get the ring 1/3 of the time: it doesn't matter what Bill does -- the chance of her winning is still 1/3.
Now let's consider what happens if she always switches:
Her initial guess will still be right 1/3 of the time. If she switches after a right guess she will never get the ring because she is always switching to a wrong box.
Her initial guess will still be wrong 2/3 of the time. If she switches after a wrong guess she will always get the ring! Why? Because Bill always opens the other wrong box, so the only remaining closed box always contains the ring.Thus, if she always switches she will get the ring 2/3 of the time.Therefore she should always switch because she is twice as likely to win by switching.
The simpler solution starts with the observation that she will get the ring 1/3 of the time if she sticks. If she switches, since there are only two possibilities (she either gets the ring or she doesn't), then she will get the ring 1 - 1/3 = 2/3 of the time. (Mathematically, probabilities must always add up to 1.)
There are lots of simulators of this game on the web. You need to play it a LOT of times (at least 100) to get a result to which you can reliably apply statistics. A simple but fast one is here. (http://forum.caithness.org/go.php?url=http://forum.caithness.org/go.php?url=http://forum.caithness.org/go.php?url=http://forum.caithness.org/go.php?url=http://www.stat.sc.edu/%7Ewest/javahtml/LetsMakeaDeal.html)
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