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**lord12** A gambling book recommends the following winning strategy for the game of roulette. It recommends that the gambler bets $\displaystyle \$1 $ on red. If red appears (which has probability $\displaystyle \frac{18}{38} $), then the gambler should take her $\displaystyle \$1 $ profit and quit. If the gamblers loses the bet (which has probability $\displaystyle \frac{20}{38} $ of occurring),

she should make additional $\displaystyle \$1 $ bets on red on each of the next new spins of the roulette wheel and then quit. Mr F asks: What does this mean? Do you mean on the next spin, the next two spins, .....??

Let $\displaystyle X $ denote the gambler's winnings when she quits.

(a) Find $\displaystyle P(X>0) $.

(b) Is this a winning strategy?

(c) Find $\displaystyle E[X] $.

For (a) $\displaystyle P(X>0) = 1 - P(X \leq 0) $. This is the easy way to compute it? Mr F says: It's one way of doing it, I suppose. But it would help to know the answer to my above question.

(b) What is the definition of a winning strategy? Mr F says: E[X] > 0.

(c) $\displaystyle E[X] = \sum x \cdot p(x) $ Mr F asks: And your question is ....?