A box is filled with 'N' slips of paper. On each slip of paper is written some positive integer (note that any positive integer may appear on the slips - not just the integers from 1 to 'N'). The integers do not necessarily appear in any sequence or pattern. Each of the slips has a different integer on it, so there is just one slip with the greatest integer.

A person who has no prior knowledge of which numbers appear on the slips - but who does know that there are 'N' slips - is to blindly pull slips from the box one by one. The person looks at each slip, then either agrees to accept that number (of Rupees) and quit or decides to go on and choose another slip.

Note that the person looks at each slip as he/she proceeds, and then decides whether to quit or to go on. That person can go forward, but cannot go back. If no choice is made by the time the 'N'th slip is reached, then the person must accept the number (of Rupees) on the 'N'th slip.

Does there " EXIST " a 'Best Strategy' for the person ? If " YES ", then what is that strategy ? (Here the term " Best Strategy" means that the person will get the greatest amount of Rupees).

If when we say random we assume any number from 1 to ∞ can be chosen. This means every number has an equal chance to be drawn. so now you draw the first slip, then you look at the number and see that it is a number, which we will call x. Because every number has an equal chance of showing up there will be x-1 possbilities you get a number lower than it next and ∞-x chances the number you will be higher than your previous slip. That means every time you should draw another slip, because of the fact that ∞-x possibilities to get a higher slip is so much more than x-1.
As said before this is under the assumption that since the number written is random every number from 1 through ∞ will have an equal chance of being drawn. If these numbers were to be chosen by a computer(which isn't truly random, its "randomness" is generated by an algorithm) then of course the numbers wouldn't have an equal chance of being written down. Same if a human were to pick random numbers out of his head, once again this is not random. The brain takes in millions of different things and this will cause electric signals in our brain to make us think of a specific number.

Since in our universe randomness does not exist, if this question were to be real life, then my answer would not apply, but following all the conditions of the questions "such as it having randomness" my solutioon is mathematically correct.

Posted by Alan on 2003-03-18 13:42:24