Side bet against a habitual roulette player

Mr. Brown always stakes $1 on number 13 at American roulette (38 equally likely pockets; a winning number returns the stake plus 35 times the stake, otherwise the stake is lost). Hoping to put him off, his friend offers a $20 even-money wager, each time, that Brown will be down after 36 spins. What is Brown's overall expected result per 36 spins, counting both the roulette and the side bet?

He gains about $2.79 per 36 spins. One hit in 36 spins leaves Brown exactly even, so he is behind only if he never hits: $(37/38)^{36}\approx0.383$. The side bet is worth $20(0.617-0.383)\approx\text{\char36}4.68$ to him, while the roulette costs $36\cdot\tfrac{2}{38}\approx\text{\char36}1.89$; the net is roughly $+\text{\char36}2.79$.