Expected value in multiple rounds

I am doing an exercise that sounds like:

The game of European roulette involves spinning a wheel with 37 slots: 18 red, 18 black, and 1 green. A ball is spun onto the wheel and will eventually land in a slot, where each slot has an equal chance of capturing the ball. Gamblers can place bets on red or black. If the ball lands on their color, they double their money. If it lands on another color, they lose their money.

First question is:

(a) Suppose you play roulette and bet $3 on a single round. What is the expected value and standard deviation of your total winnings? I created this table to solve it: Where I got SD = 2.14 and Expected value = -0.09 However the second question I don't know what to do: (b) Suppose you bet$1 in three different rounds. What is the expected value and standard deviation of your total winnings?

Can I just do E(X) * 3?

1 Answer

Guide:

Number the rounds $$1,2,3$$ and write: $$X=X_1+X_2+X_3$$ where $$X_i$$ denotes the winning in round $$i$$ for $$i=1,2,3$$.

Apply linearity of expectation to find $$\mathbb EX$$.

Realize that the $$X_i$$ are independent and use the rule: $$\mathsf{Var}(U+V)=\mathsf{Var}(U)+\mathsf{Var}(V)$$ for independent random variables $$U,V$$ for which variance is defined.