Uninformed Spread Profit

Compute the market maker's profit separately for a buy trade and a sell trade against fair value.

Average the two profits since each is equally likely.

1. 1. Profit on each side

   If the trader buys at the ask 101, the market maker's profit versus fair value 100 is $101-100=1$. If the trader sells at the bid 99, the market maker's profit is $100-99=1$.

2. 2. Average

   Both outcomes give a profit of 1, and each occurs with probability $1/2$, so the expected profit is $0.5(1)+0.5(1)=1$.

   $$E[\text{profit}]=0.5(1)+0.5(1)=1$$

Answer: 1