roll a number cube. if the number cube comes up odd, you win the same number of points as the number on the…

Explanation:

Step1: Identify possible outcomes and probabilities

A number - cube has 6 faces numbered 1 - 6. The probability of getting an odd number ($P(O)$) is $\frac{3}{6}=\frac{1}{2}$, and the probability of getting an even number ($P(E)$) is $\frac{3}{6}=\frac{1}{2}$. The odd numbers are 1, 3, 5 and the even numbers are 2, 4, 6.

Step2: Calculate the expected value for odd - number outcomes

For odd - number outcomes: When the number is 1, the points are 1; when the number is 3, the points are 3; when the number is 5, the points are 5. The expected value for odd - number outcomes $E(O)$ is $\frac{1\times1 + 3\times1+5\times1}{6}=\frac{1 + 3+5}{6}=\frac{9}{6}=\frac{3}{2}$.

Step3: Calculate the expected value for even - number outcomes

For even - number outcomes, when the number is even, you lose 4 points. So the expected value for even - number outcomes $E(E)$ is $\frac{(- 4)\times3}{6}=-2$.

Step4: Calculate the overall expected value

The expected value $E(X)$ of a roll is $E(X)=E(O)+E(E)$. Using the values from Step 2 and Step 3, $E(X)=\frac{3}{2}+(-2)=\frac{3 - 4}{2}=-\frac{1}{2}=-0.5$.

Answer:

-0.5