Question

roll a number cube. if the number cube comes up odd, you win the same number of points as the number on the cube. if the number comes up even, you lose 4 points. what is the expected number of points per roll? -0.25 -0.5 0 0.25 0.5 done

Explanation:

Step1: Determine probabilities and values

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

Step2: Calculate expected value for odd numbers

The expected - value contribution from odd numbers is $P(O)\times(1 + 3+5)\div3$. Since $P(O)=\frac{1}{2}$ and $(1 + 3+5)\div3 = 3$, the contribution is $\frac{1}{2}\times3=\frac{3}{2}$.

Step3: Calculate expected value for even numbers

The expected - value contribution from even numbers is $P(E)\times(- 4)$. Since $P(E)=\frac{1}{2}$, the contribution is $\frac{1}{2}\times(-4)=-2$.

Step4: Calculate total expected value

The total expected value $E$ is the sum of the expected values from odd and even numbers. $E=\frac{3}{2}+(-2)=\frac{3 - 4}{2}=-\frac{1}{2}=-0.5$.

Answer:

-0.5