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

Explanation:

Step1: Determine probabilities and outcomes

The sample - space of rolling a number cube is \(S = \{1,2,3,4,5,6\}\), \(n(S)=6\). The odd numbers are \(1\), \(3\), \(5\) and the even numbers are \(2\), \(4\), \(6\). The probability of getting an odd number \(P(\text{odd})=\frac{3}{6}=\frac{1}{2}\), and the probability of getting an even number \(P(\text{even})=\frac{3}{6}=\frac{1}{2}\).

Step2: Calculate the expected - value formula

The expected - value formula is \(E(X)=\sum_{i}x_ip_i\), where \(x_i\) are the possible outcomes and \(p_i\) are their corresponding probabilities. For odd numbers: when \(x = 1\), \(p=\frac{1}{6}\); when \(x = 3\), \(p=\frac{1}{6}\); when \(x = 5\), \(p=\frac{1}{6}\); for even numbers, when \(x=-4\) (since losing 4 points), \(p=\frac{3}{6}\).
\[

$$\begin{align*} E(X)&=(1\times\frac{1}{6})+(3\times\frac{1}{6})+(5\times\frac{1}{6})+(- 4\times\frac{3}{6})\\ &=\frac{1 + 3+5-12}{6}\\ &=\frac{9 - 12}{6}\\ &=-\frac{3}{6}\\ &=-0.5 \end{align*}$$

\]

Answer:

-0.5