Steven and I have a biased coin. The coin has a 90% chance to show heads, and a 10% chance to show tails. We flip the coin in the same way 4 times. Steven picks heads for the first two rounds and I pick tails for the first two rounds. Then I choose heads for the last two rounds and Steven chooses tails for the last two rounds.
In order to win, one of us must succeed 3 times. Is one of us more likely to win?
My calculation shows that we each have a 0.1557 chance of winning. My friend is arguing that I am wrong based on absorbing markov chains.
Here's what I've done:
Am I crazy or is this really obvious that it doesn't matter which side you start on?