A man tossing a coin wins one point for heads and five points for tails. The game stops when the man accumulates at least 1000 points.

How do you estimate with accuracy the expectation of the length of the game?

A first observation is that the game necessarily stops between 200 and 1000 tosses. Those numbers being "large" enough, the expectation being 3 and the variance 4, e.g. finite, the Law of large Numbers tells us that a first estimate of the average length is 334.

But obviously it is possible to do better than that. (sub)Martingale theory?

Thanks for your help.