1. ## Roulette

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 $bets on red on each of the next new spins of the roulette wheel and then quit. Let$\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) $. This is the easy way to compute it? (b) What is the definition of a winning strategy? (c)$\displaystyle E[X] = \sum x \cdot p(x) $2. Originally Posted by lord12 A gambling book recommends the following winning strategy for the game of roulette. It recommends that the gambler bets$\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 ....?
..

3. next two spins.

4. Originally Posted by lord12
next two spins.
Then the only possible value of X that's greater than zero is X = 1. This occurs for either LWW or W. So it's not difficult to calculate Pr(X = 1) ....

The other possible values of X are:

X = -3 which occurs for LLL

X = -1 which occurs for LWL or LLW.

So it's not difficult to calculate Pr(X = -3) and Pr(X = -1).