You are offered following game to play: a fair coin is tossed till heads turns up for the first time. If this takes place on the first toss you receive 2 dollars, if it takes place on the second toss you receive 22=4 dollars and, in general, if heads turns up for the first time on the nth toss you receive 2n dollars.
(a) show that the expected value of your winnings does not exist (i.e., is given by a divergent sum) for this game. Does this mean that this game is favorable no matter how much you pay to play it?
(b) Assume that you only receive 210 dollars if any number greater than or equal to ten tosses are required to obtain the first head. Show that your expected value for this modified game is finite and find its value.