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Sat 2 Nov, 2013 04:40 pm
Hi, group,
Semi-retiree here with newfound interest in probability, slowly working my way through the problems in Wm Feller's excellent book. Problem 11 in section II.10 is: A man is given n keys of which only one fits his door. He tries them successively (sampling without replacement). This procedure may require 1, 2, ..., n trials. Show that each of these n outcomes has probability 1/n.
Feller's answer: The probability of exactly r trials is P( n–1, r–1 )/P( n, r ) = 1/n, where P( n, r ) = number of permutations of r objects taken from n objects (his notation is different but hard to type in a forum post).
But I can reason like this: since each of the n keys has an equal probability = 1/n to be the right key, the probability that the man's procedure will terminate on the first, second, ..., r-th key also has to be 1/n. The problem is sort of trivial.
I guess I don't know what this problem was designed to teach. I can derive Feller's answer using a probability tree, but don't really know what lesson I'm meant to take away. Comments?
@X-man,
On the first try, there is a 1/n chance of the key being correct. If it is not, on the second try, there is a 1/(n-1) chance of the key being correct. And so on. Of course, you only get to the second try with probability (n-1)/n. Maybe the lesson is that there's more than one way to skin a cat (often the case with counting (permutations, combinations, etc.) and probability.
@markr,
Thanks, markr. By the way, it probably is a typo, but Feller's answer isn't correct. P( n–1, r–1 )/P( n, r ) is NOT = to 1/n (the equality doesn't hold).
@X-man,
Sure it does.
P(n-1,r-1) = (n-1) * (n-2) * ... * (n-r+1)
P(n,r) = n * (n-1) * (n-2) * ... * (n-r+1)
@markr,
Well, when you put it that way, markr, of course it does
I don' t know how I missed that. I guess I was trying to concentrate on Feller's thought process. The way he chooses to give the answers to his problems sort of gives hints as to how to arrive at the solution. I kept trying to think of permutations, permutations when solving this problem.
But thanks again. You may want to look at a Feller problem I posted on another forum. I couldn't do it here because I couldn't find a way to upload pdf and xls files connected with it.
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