Wednesday, June 6, 2007

Bayesian Puzzle

Today's game shows offer a lot. Lets work around what to choose. The well known Bayesian Puzzler
Its so mind boggling when we have to choose from what we have and what we can have that is unknown.
Ever wondered when you are being given an option to choose from the given to things. Of which you know one contains half the sum of money then the other one. But you don't know which envelope contains what. Now when you choose one of them and finds out it contains a 100 units of money. And now you are being given another option whether to choose the other envelope or take away the one you have opened. Now what?
Would you go for it??
Now looking over the situation one find 50% chances for both the situations and accordingly it wont matter if you switch or not. But it isn't the way as it looks like.
Think this way. Now say you have 100 units in the envelope you opened. The other one will either have 200 or 50. Correct?
Now there is a 50% chance for both outcomes. If you dig deeper in terms of weightage of those chances it would be 200*0.5 and 50* 0.5. That means if you choose to go for the second envelope it would be 200*0.5 + 50* 0.5 =125. That means at the cost of 100 units you will get 125. And so is the net profit of 25 units. So now it looks , it always make sense to switch to the second envelope, isn't it??

Now look at the problem with a different perspective. The above logic says, even if you don't look inside the envelopes , you must switch. Yes you should, because it would be something like
You got x units then on choosing the other envelope it will be 0.5*(2X) + 0.5*(0.5X) = 1.25X.
So there will always be a 25% gain. So you must switch.

You see there is not much to think about. Next time you indulge in some situation/riddle like this, you have the answer to yourself , what to choose!!!