Chapter 4, Problem 4.20P

A gambling book recommends the following “winning strategy” for the game of roulette: Bet $1 on red. If red appears (which has probability $\frac{18}{38}$ ) then take the $1 profit and quit. If red does not appear and you lose this bet (which has probability $\frac{20}{38}$ of occurring), make additional $1 bets on red on each of the next two spins of the roulette wheel and then quit. Let X denote your winnings when you quit.

a. Find $P\left\{X>0\right\}$.

b. Are you convinced that the strategy is indeed a “winning” strategy? Explain your answer!

c. Find $E\left[X\right]$.