Diminishing Hopes

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.

There are several questions:

The fellow checked and found no documents in the first drawer. What is the probability that the document will be found in the remaining seven drawers?
The fellow checked and found no documents in the first four drawers. What is the probability that the document will be found in the remaining four drawers?
The fellow checked and found no documents in the first seven drawers. What is the probability that the document will be found in the last remaining drawer?

Answers

Solution

The answers to the three questions are 7/9, 2/3 and 1/3, respectively. The result may be surprising. Chances of locating the document decrease with every checked drawer. The general formula for the probability of finding the document after unsuccessfully checking n drawers is P_n = (8 - n)/(10 - n). This is a decreasing function of n for n < 10. You can check the solution.

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 P_n = (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 Q_n = 1/(10 - n), an increasing function of n for n < 10.

References

Ruma Falk, Understanding Probability and Statistics, A K Peters, 1993

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