Question

At a County fair there is a spinner game with 12 sectors. 2 red sector, 2 green sector, 2 blue sector, and 6 yellow sectors. If the spinner lands on a red sector, the player wins 2 tokens. If the spinner lands on a green sector, the player wins 2 tokens. If the spinner lands on a blue sector, the player wins 2 tokens. If the spinner lands on a yellow sector, the player loses 3 tokens. Is this game fair for the player and how much will the player win or lose on an average over time?

Answer 1

Let the token profit be X.
For a win, the profit is 2 tokens.
For a loss, the profit is -3 tokens.
The probability of a win is
\[\frac{2}{12}+\frac{2}{12}+\frac{2}{12}=\frac{6}{12}=\frac{1}{2}\]
the probability of a loss is
\[\frac{6}{12}=\frac{1}{2}\]
Setting up a probability distribution variable gives:
\[E(X)=(-3\times \frac{1}{2})+(2\times \frac{1}{2})=-1\frac{1}{2}+1=-\frac{1}{2}\ token\]
Therefore, on average, the player would expect to lose a half a token per game.