probability and expected value Hey I am not sure if I thinking correctly on this question? In a carnival, there is game which charges you $3$ dollars to play a game. You win $1$ dollar for every consecutive head you get and you you can play till you get tail. if you get head head tail you get back two dollars. What is expected profit?
my theory is your probability of t,ht,hht,hhht and so on will be $1/2,1/4,1/8,1/16$ and so on, and income will be $0,1,2,3$ and so on.
so average income will be 
$$ 0 * \frac{1}{2} + 1 * \frac{1}{4} + 2 * \frac{1}{8} + 3 *\frac{1}{16} + ....$$
which will come to a dollar?
so average profit will be, $1-3 = -2$??

Taking "for every consecutive head" to mean "number of heads before a tail", your formulation is correct, and here is another way to get its sum.
A = $0\times\frac{1}{2} + 1\times\frac{1}{4} + 2\times\frac{1}{8} + 3\times\frac{1}{16} + ...$
$\frac{A}{2}$ = ............ $0\times\frac{1}{4} + 1\times\frac{1}{8} + 2\times\frac{1}{16} + ...$
Subtracting, $\frac{A}{2} = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ...$ , a well known series with sum = 1/2
thus expected net gain = 1 - 3 = -2
