1. Sherlock and Mycroft play a game which involves flipping a single fair coin. The coin is flipped repeatedly until one person wins. Sherlock wins if the sequence TTT (tails-tails-tails) shows up first while Mycroft wins if the sequence HTT(heads-tails-tails) shows up first. Who among the two has a higher probability of winning?
1. Sherlock has probability $1 / 8$ of winning while Mycroft has probability greater than $\frac{1}{8}$. The event "Sherlock wins" is just the set $\{T T T\}$ so that $P(\{T T T\})=\frac{1}{8}$ while the event "Mycroft wins" is the set $M=$ $\{H T T, H H T T, T H T T, H H H T T, T T H T T, H T H T T, T H H T T, \cdots\}$ and $P(M)>\frac{1}{8}$.