[mannika]

I ran a simulation and came up with answers of:
Win all his money 94.67% of the time (st.dev of 0.225%) with $1 wagers.
Win all his money 99.77% of the time (st.dev of 0.048%) with $0.50 wagers.

(Sorry, I don't know how to do the math, and therefore cannot help you out with that part)

//procrastinating studying for derivative securities final