given the spinner below in which all regions are equal, which of the following point scales would result in…

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

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

Answer

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.