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?

Question

Accepted Answer

# Explanation:
## Step1: Determine probabilities and values
A standard number cube has 6 faces: 1, 2, 3, 4, 5, 6.
Odd numbers: 1, 3, 5. Probability of odd $P(\text{odd})=\frac{3}{6}=\frac{1}{2}$.
Values for odd: $x_1 = 1$, $x_2=3$, $x_3 = 5$.
Even numbers: 2, 4, 6. Probability of even $P(\text{even})=\frac{3}{6}=\frac{1}{2}$. Value for even $x = - 4$.
## Step2: Calculate expected value formula
The expected - value formula is $E(X)=\sum_{i}x_ip_i$.
For odd numbers:
$$
\begin{align*}
E(X_{\text{odd}})&=(1\times\frac{1}{6})+(3\times\frac{1}{6})+(5\times\frac{1}{6})\
&=\frac{1 + 3+5}{6}\
&=\frac{9}{6}=\frac{3}{2}
\end{align*}
$$
For even numbers: $E(X_{\text{even}})=(-4)\times\frac{1}{2}=-2$
## Step3: Calculate total expected value
$E(X)=E(X_{\text{odd}})+E(X_{\text{even}})$
$$
\begin{align*}
E(X)&=\frac{3}{2}+(-2)\
&=\frac{3 - 4}{2}\
&=-\frac{1}{2}=- 0.5
\end{align*}
$$

# Answer:
$-0.5$

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QUESTION IMAGE

Question

Explanation:

Step1: Determine probabilities and values

Step2: Calculate expected value formula

Step3: Calculate total expected value

Answer: