A fellow's desk sports 8 drawers, where he randomly (but with equal probabilities) stores his documents. This would be the whole condition of the problem, except that in 2 out of 10 cases the fellow simply forgets to store a document, and eventually the latter gets lost.
When the fellow needs a document he starts a search from the first drawer and proceeds sequentially until the document is found or it becomes clear (after checking all the drawers) that the document has not been stored in the desk in the first place.
Let's add two imaginary drawers to the desk and assume that this is where the documents get lost. We may even think of them as being locked during a search.
With K drawers to go, of which two are unavailable, the probability of a successful outcome is (K - 2)/K. With K = 10 - n, this reduces to Pn = (8 - n)/(10 - n). This is a decreasing function of n for n < 10.
Another curiosity: if we only ask about the probability of finding the document in the next available drawer (after checking the first n drawers), the probability will be Qn = 1/(10 - n), an increasing function of n for n < 10.
References
- Ruma Falk, Understanding Probability and Statistics, A K Peters, 1993
Copyright © 1996-2009 Alexander Bogomolny