[Math] Expected gain or loss in roulette

expectationprobabilityprobability theory

The questions reads:

There are $37$ numbers, from $0$ to $36$. Each number has an equal chance of turning up. Zero is green in color and odd numbers are in black and even numbers are in red. If you place $\$1$ on red (black) you get $2$ dollars if you are right and $\$0$ if you are wrong. For each bet there are $19$ losing numbers and $18$ winning numbers.

Suppose you place $\$1$ bets $18$ times on red. What is your expected gain or loss after $18$ tries?

My working:

Expected value of one game = $2(18/37) + -$1(19/37)

Expected value of 18 games = ($2(18/37) + -$1(19/37))^18

Is this right?

Part 2:

Suppose I use a dollar for each color bet? How many games can I play if I have an initial capital of $18?

Not sure how to start on this…

Best Answer

It's the right way to go about it, but there are two bugs:

  • The value of a win is $1$, not $2$.
  • The expected outcome after $18$ rounds is $18$ times the expected expected outcome after $1$ round, by linearity of expectation.
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