I'm working out of Feller's "Introduction to Probability and its Application (Vol I.)" textbook and I'm stuck on a coin toss problem. I'll list the full problem and show where I'm having trouble.
A coin is tossed until for the first time the same result appears twice in succession. To every possible outcome requiring n tosses attribute probability 1/$2^{n-1}$. Describe the sample space. Find the probability of the following events: a.) the experiment ends before the sixth toss, b.) an even number of tosses is required.
Alright so I'm not having any trouble describing the sample space and completing part a. This first part was solved by creating a possibility tree and adding up the probabilities (answer: 15/16). However, I'm stuck on part b and I don't understand how the 1/$2^{n-1}$ given in the problem is to be interpreted because if you toss the coin twice it makes it seem like HH, and TT each have a probability of 1/2 which is not the case. The sample space of two tosses would be {HH, HT, TH, TT} and each would have a probability of 1/4 and following this logic I arrived at 15/16 so I believe this is the correct thinking which makes the problem even more confusing.
The answer to part b is 2/3 so I'm not sure if that will help. Thanks for any help.