Given:
A standard 6-sided dice is rolled.
If you roll an even number you get two points.
If you roll an odd number you lose one point.
To find:
The expected number of points per roll?
Solution:
If a dice is rolled, then the possible outcomes are 1, 2, 3, 4, 5, 6.
Odd values are 1, 3, 5 and the even values are 2, 4, 6.
The probability of getting an odd number is:
![P(odd)=\dfrac{3}{6}](https://tex.z-dn.net/?f=P%28odd%29%3D%5Cdfrac%7B3%7D%7B6%7D)
![P(odd)=\dfrac{1}{6}](https://tex.z-dn.net/?f=P%28odd%29%3D%5Cdfrac%7B1%7D%7B6%7D)
The probability of getting an even number is:
![P(even)=\dfrac{3}{6}](https://tex.z-dn.net/?f=P%28even%29%3D%5Cdfrac%7B3%7D%7B6%7D)
![P(even)=\dfrac{1}{6}](https://tex.z-dn.net/?f=P%28even%29%3D%5Cdfrac%7B1%7D%7B6%7D)
The expected number of points per roll is:
![E(x)=2\times P(even)-1\times P(odd)](https://tex.z-dn.net/?f=E%28x%29%3D2%5Ctimes%20P%28even%29-1%5Ctimes%20P%28odd%29)
![E(x)=2\times \dfrac{1}{2}-1\times \dfrac{1}{2}](https://tex.z-dn.net/?f=E%28x%29%3D2%5Ctimes%20%5Cdfrac%7B1%7D%7B2%7D-1%5Ctimes%20%5Cdfrac%7B1%7D%7B2%7D)
![E(x)=1-0.5](https://tex.z-dn.net/?f=E%28x%29%3D1-0.5)
![E(x)=0.5](https://tex.z-dn.net/?f=E%28x%29%3D0.5)
Therefore, the expected number of points per roll is 0.5.