TITLE: Calculating average win Wheel of fortune, with possible re-spins.
QUESTION [0 upvotes]: I'm working on a wheel of fortune, there are $10$ prize slots and the winning slot is picked randomly:
$7 \times 100$ points
$2 \times 0$ points
$1 \times +3$ re-spins.
So you have a $10\%$ chance to win $3$ extra spins, $20\%$ to win $0$ points, and $70\%$ to win $100$ points.
I would like to know how to calculate the average win you would get with one spin, including the possible winnings gained from re-spins.
Please include in your response a formula which is readable by a layman. (I'm no mathematician)

REPLY [1 votes]: The average (or expected) value given by a spin is found by multiplying each possible outcome by its probability of occuring. We write the expected value as $E(X)$ where $X$ is the variable representing the value obtained in a single spin. We then have
$$E(X)=\frac{70}{100}\times100+\frac{20}{100}\times0+\frac{10}{100}\times(3\times E(X))$$
where we have a seventy percent chance of obtaining $100$ points, twenty percent chance of $0$ points and ten percent chance of getting three times the expected value (on average). Hence rearranging the above equation, the average points scored in one spin is
$$E(X)=100$$
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