The expected value is the summary of the product between each element and its probability.
In this case, we have to multiply the number of dollars in each area with the probability of happening. It's important to note that there are 8 portions inside the circle because each of them has 45°.
According to the given diagram $160 represents one slice, $80 represents one slice, and -$40 represents 6 slices.
Having said that, we can express the expected value as follows.
[tex]\begin{gathered} E=160\cdot(\frac{1}{8})+80\cdot(\frac{1}{8})-40\cdot(\frac{6}{8}) \\ E=\frac{160}{8}+\frac{80}{8}-\frac{240}{8}=20+10-60=-30 \end{gathered}[/tex]
Therefore, the expected value is -30 which means you most likely will lose $30 if you play.
