**Abstract**

**The****My project deals with an idealized but still realistic situation described as follows. A finite number of offers arrive one at a time. If an offer is accepted, the search ends. All rejected offers vanish. The amount of offers is known. The offers are characterized by a numeric value (e.g. price), but the range of this value is unknown. The goal of my project is to find an efficient strategy for making a good choice in such a situation and to estimate the chances of getting the top offer with the help of this strategy.**

*objective:*** Methods/Materials**

I reasoned that a good strategy is to skip the first few offers and then to choose the first offer which beats all that were skipped. To find how many offers should be skipped, I used both theoretical and experimental methods. In the theoretical part, I computed the probability of selecting the best of N offers after skipping S offers. I found a formula, evaluated it for different N and S, and, analyzing the obtained tables and graphs, found the best amount of offers to skip. In the experimental part, I verified the prediction by simulating a big number of searches. To imitate each search, I generated N random numbers (I used N=10 and N=50) which were considered as offers, and "ran" several searches skipping different amounts of offers. For reliability, every experiment was repeated 100 times with different random numbers.

** Results**

The best strategy is to skip about 37% of the total amount of offers and then pick the first offer which beats all of the skipped offers. Amazingly, this recipe gives about 37% chances of getting the top offer, and this predicted rate of success does not decrease when the number of offers increases.

** Conclusions/Discussion**

My project shows that knowledge and reasonable patience without overcautiousness secures about 37% of success in the search with no additional information about the options. This method may be used in a different situation as well. For example, if you have to make a choice before a deadline, be patient and spend 37% of the available time plainly "observing" the situation.

This Mathematical project is to choose the best from a given number of offers, is recommended to skip the first 37% of all the offers and then to pick up the first offer topping all its predecessors --- this strategy gives about 37% of success!

**Science Fair Project done By Nina Budaeva **