Down Two Before Up Three

You and a friend flip a fair coin for $\$1$ a flip, and you track your net winnings starting from $0$. You agree in advance to quit the moment you are either down $\$2$ or up $\$3$. What is the probability the session ends with you down $\$2$? Would the answer change if your running P&L instead followed a driftless Brownian motion stopped at the same two levels?