Out of all the people you could possibly date, see about the first 37%, and then settle for the first person after that who's better than the ones you saw before (or wait for the very last one if such a person doesn't turn up). You don't want to go for the very first person who comes along, even if they are great, because someone better might turn up later. Either way, we assume there’s a pool of people out there from which you are choosing.On the other hand, you don't want to be too choosy: once you have rejected someone, you most likely won't get them back. And since the order in which you date people might depend on a whole range of complicated factors we can’t possibly figure out, we might as well assume that it’s random.
Your strategy is to date of the people and then settle with the next person who is better.Our task is to show that the best value of corresponds to 37% of .We’ll do that by calculating the probability of landing X with your strategy, and then finding the value of that maximises this probability.In other words, you pick X if the highest-ranked among the first people turned up within the first people. In other words, you pick X if the highest-ranked among the first people turned up within the first people. If , so there are only five people, the only value of for which the two inequalities hold is , which is 40% of : So you should discard the first two people and then go for the next one that tops the previous ones.These percentages are nowhere near 37, but as you crank up the value of , they get closer to the magic number.For twenty potential partners () you should choose , which is 35% of . For a hundred potential partners () you should choose (that’s obviously 37% of ) and for (an admittedly unrealistic) 1000 () you should choose , which is 36.8% of .