Question

given the spinner below in which all regions are equal, which of the following point scales would result in a favorable game? the odd numbered regions will add one point. the even numbered regions lose one point. red will add one point. any other color will lose one point. a number less than or equal to 3 will add one point. a 4 will lose two points

Explanation:

Step1: Calculate expected value for first option

There are 2 odd - numbered regions (1 and 3) and 2 even - numbered regions (2 and 4). Probability of landing on odd or even is $P(\text{odd})=P(\text{even})=\frac{2}{4}=\frac{1}{2}$. Expected value $E_1=\frac{1}{2}\times1+\frac{1}{2}\times(- 1)=0$.

Step2: Calculate expected value for second option

There is 1 red region and 3 non - red regions. Probability of landing on red is $P(\text{red})=\frac{1}{4}$ and on non - red is $P(\text{non - red})=\frac{3}{4}$. Expected value $E_2=\frac{1}{4}\times1+\frac{3}{4}\times(-1)=\frac{1 - 3}{4}=-\frac{1}{2}$.

Step3: Calculate expected value for third option

There are 3 numbers less than or equal to 3 (1, 2, 3) and 1 number equal to 4. Probability of landing on a number $\leq3$ is $P(\leq3)=\frac{3}{4}$ and on 4 is $P(4)=\frac{1}{4}$. Expected value $E_3=\frac{3}{4}\times1+\frac{1}{4}\times(-2)=\frac{3 - 2}{4}=\frac{1}{4}>0$.

Answer:

A number less than or equal to 3 will add one point. A 4 will lose two points.