5 tosses of a coin are made We receive 1 dollar for every coin toss, up to and including the first time a head comes up. Then, we receive 2 dollars for every coin toss, up to the second time, a head comes up. More generally, the dollar amount per toss is doubled each time a head comes up.
What will the sample space be and how should I calculate it if I want to calculate the probability that I will make 10 dollars?
I think that in this game there will be only one unique way to win a certain dollar amount (I am unable to prove this concretely) For example, the only way to win 7 dollars is T T H T H , the only way to make 6 dollars is T T T H H .
I was thinking that the sample space would be A = P(5,1) + P(5,2) ... + P(5,5) . This will account for all the ways in which heads can occur in 5 tosses. Since I think there is only one unique way to get a certain dollar amount, the probability would be 1 /A.
Is my way of thinking correct?