Question
Bob proposes a game where Alice and Bob are each given a coin. Each is allowed decide the probability of heads for their coin. Afterwards, they both flip their coins. If it comes up $HH$, Alice gives Bob $\$6$. If it comes up $TT$, Alice gives Bob $\$4$. Otherwise, Bob gives Alice $\$5$. However, Alice must tell Bob the success probability $p_1$ they have chosen before Bob decides his $p_2$. The interval of values for $p_1$ such that regardless of what Bob selects, Bob has non-positive expected value on the game is in the form $[a,b]$, where $a$ and $b$ are rational numbers. Find $b-a$.