[allenz]

You might find the result less surprising after you solve a riddle by Martin Gardner:

> A young man lives in Manhattan near a subway express station. He has two girlfriends, one in Brooklyn, one in the Bronx. To visit the girl in Brooklyn, he takes a train on the downtown side of the platform; to visit the girl in the Bronx, he takes a train on the uptown side of the same platform. Since he likes both girls equally well, he simply takes the first train that comes along. In this way, he lets chance determine whether he rides to the Bronx or to Brooklyn. The young man reaches the subway platform at a random moment each Saturday afternoon. Brooklyn and Bronx trains arrive at the station equally often—every 10 minutes. Yet for some obscure reason he finds himself spending most of his time with the girl in Brooklyn: in fact on the average he goes there 9 times out of 10. Can you think of a good reason why the odds so heavily favor Brooklyn?

The idea shows up again in the Elevator paradox, which has a delightful article on Wikipedia: https://en.wikipedia.org/wiki/Elevator_paradox

[tgb]

I see an immediate solution to that riddle and it matches the idea of the Wikipedia page you link. But I don't see any connection to Penney's game. Can you explain?

[frumiousirc]

The operative words in the rules of both riddles are "appears first".

[allenz]

Yup, Penney's game is surprising because at first glance the probabilities of two sequences of the same length are unrelated. But if more than half of the sequences overlap, then one sequence will tend to arrive before the other. As the proportion of overlap tends to 100%, Player B has a 2:1 advantage over Player A: https://i.imgur.com/eKujwrK.png