Make Me a Market: The Bid-Ask Game

You are a market maker. Your best estimate of a stock's value is $V = \$50$ with uncertainty: the true value is uniformly distributed on $[40, 60]$. You must quote a bid $B$ and ask $A$ (with $B < A$). Traders who arrive are either informed (probability $\alpha = 0.3$, they know the true value and only trade if profitable for them) or uninformed (probability $0.7$, equally likely to buy or sell regardless of true value). (a) If you set $B = 48$ and $A = 52$, what is your expected profit per trade? (b) What is the break-even spread?