Imagine the following experience:
You toss a coin until I get n heads OR n tails. That means that the maximum amount of tosses I will perform is 2n-1. I now want to find the probability that I obtain n heads first, given that the first toin coss was a heads.
I tried: P(obtain n heads | first is heads) = P(obtain n heads AND first is heads)/P(first is heads).
The denominator is one half, since the coin is fair. Therefore, the probability I am looking for is 2P(obtain n heads AND first is heads). I've now been trying to count the cardinality of the two sets A and B such that A is the set of all possible games that could have been played in which I obtain n heads and started with a head, and B is the set of all possible games that could have been played total.
My approach to count A: Consider a multiset with n-2 heads (two heads have been used up for the first and last game) and i tails. We can now take the sum from i=0 to n-1 of the permutation of that multiset.
Same for B, but we consider the multiset with n-1 heads and i tails. (one head has been used up as the first game)
Divide the cardinality of A by the cardinality of B. Multiply the result by 2. This is my current result to the problem.
However, I'm pretty sure that it is wrong, as I wrote a simulation to this problem that gives me a different probability. Any tips?