Quoting Against Informed Flow

An asset is worth $X \sim \text{Uniform}(0,1)$, drawn only after you quote. You post a bid $b$ and an ask $a$, and 1000 traders arrive:

• 300 are informed: each knows $X$ exactly.
• 700 are uninformed: each draws an independent guess $G \sim \text{Uniform}(0,1)$ and treats it as the value.

Every trader trades at most one unit and only when they expect a profit, so a trader buys from you at $a$ when their value estimate exceeds $a$, and sells to you at $b$ when it falls below $b$. Choose $a$ and $b$ to maximise your expected profit.

What is the resulting spread $a - b$?

Answer format: round to 3 decimal places.