Your goal is to select a puppy, given the following constraints.

• Sample Size - There are N puppies available for adoption.

• Random Order - The N puppies will be presented to you in a random order.

• Instantaneous Choice - You see each puppy once and only once, and must decide at that time whether or not to adopt her. If you decide not to adopt a given puppy, she goes immediately off to a home somewhere else. If you do decide to adopt a given puppy, you will not see any more candidates.

Let us first consider the case of an infinite number of puppies. A good working strategy (a wait and see algorithm) is to look at enough puppies to develop a sense of the range available, and then to select the first puppy which exceeds this range. We assume that puppy qualities can be combined into a single quantity, and that all N candidates could be sorted from 1 (least acceptable) to N (best match to your requirements).

Restating this mathematically, for a finite number of puppies, we first observe puppies. The best will be ranked , where puppy ranks can range from 1 to N. We then select the next puppy which is ranked higher than . Our goal is to determine a best value for .

We define C_m() as the event that the largest number drawn from the first m draws will appear in the first draws.

Let X be the draw of the best suited puppy. We require that two conditions be met, in order to select this candidate.

The puppy with the largest rank which is drawn before the best puppy, X, is drawn must lie within the survey group, not after it but before X. We thus require {X = + j + 1} and C_+j(), for 0 j < N. We can restate the first condition as follows.

The total condition is then

The probability of making the correction decision is

We can convert the sum into an integral, and use a simple variable substitution to solve it.

We maximize P[D] where the slope is unity with regards to , and use the chain rule and integration by parts to determine

Therefore, we maximize the probability of selecting the puppy best suited to our requirements if we select the best one that we see after first surveying roughly one-third of the pool.