Consider an infinite sequence of $1$'s and $0$'s such as the following:
$$1000111100011100000101010101010101.........$$
$1$ appears with a probability of $0.5$ and so does $0$
Player $1$ wins if $0011$ appears
Player $2$ wins if $1111$ appears
If one player wins, the game stops so only one player can win
do players $1$ and $2$ have the same chance of winning?
My work:
The immediate answer one would think of is that both players have the same chance of winning given that if taken independently of the infinite sequence both $0011$ and $1111$ have a probability of occurring equal to ${1}\over {16}$ However this answer must surely be wrong as a simple computer program simulating the experiment seems to indicate that: Player $1$ wins with probability $3\over 4$ Player $2$ wins with probability $1\over 4$ I am having trouble representing this situation mathematically however given that the winning subsequences can appear anywhere and at anytime. Although this explanation is very muddled, I mean to say that a simple probability tree is hard to use. Perhaps markov chains would be better$?$