Bid-Ask Spread and Market Making

Problem

A market maker quotes a bid of $49.90 and an ask of $50.10 for a stock.

1. What is the bid-ask spread?
2. If the market maker buys 100 shares at the bid and sells 100 shares at the ask, what is their profit?
3. If the market maker believes the true value is $50.00, what is their expected profit per trade against an uninformed counterparty?
4. Why might a market maker widen the spread?

This problem covers fundamental concepts that every quant trader is expected to know.