7. Consider the following game and betting strategy. The game is you bet $k, and a fair coin is flipped. If it comes up heads, you win 2k dollars, if it comes up tails, you lose your k dollars (i.e. if you will you gain k, if you lose you gain -k). The expected gain is k() + (k) () = 0, so it is a fair game. Here is your strategy: You bet $1 on the first round, $2 on the second round, $4 on the third round,, $21 on the nth round. You keep going until you win, then you walk away. If you win on the first round you walk away with $1, if you win on the second round you walk away with -$1+ $2 = $1. Show that if you win on the nth round you walk away with $1. What is wrong with this? It appears you always get $1. So if M your net gain, then E[M] = 1. But the game was supposed to be fair! It sounds great, all we have to do is find a game sort of like this (they exist) and start playing! Of course, we need to have quite a lot of money going in, or it is not going to work: Obviously if we start with $1, we either go away with $2 or $0 dollars and it didn't work the way we hoped. Suppose we start with $N and N is huge, like a billion dollars (we told Elon Musk our trick and he lent us the money). Of course, there is some tiny probability we just keep not winning and after a certain amount of time we have to leave the game because we just ran out of money. How small is this probability? Calculate the expected net gain with this new constraint (reality).