marleny is creating a game of chance for her family. she has 5 different colored marbles in a bag: blue…

marleny is creating a game of chance for her family. she has 5 different colored marbles in a bag: blue, red, yellow, white, and black. she decided that blue is the winning color. if a player chooses any other color, they lose 2 points. how many points should the blue marble be worth for the game to be fair?\n4\n6\n8\n10

marleny is creating a game of chance for her family. she has 5 different colored marbles in a bag: blue, red, yellow, white, and black. she decided that blue is the winning color. if a player chooses any other color, they lose 2 points. how many points should the blue marble be worth for the game to be fair?\n4\n6\n8\n10

Answer

Explanation:

Step1: Calculate probabilities

The probability of picking the blue marble is $P(\text{blue})=\frac{1}{5}$, and the probability of picking a non - blue marble is $P(\text{non - blue})=\frac{4}{5}$.

Step2: Set up expected - value equation

Let $x$ be the number of points for picking the blue marble. For a fair game, the expected value $E(X)$ is 0. The value for picking a non - blue marble is $- 2$ points. So, $E(X)=P(\text{blue})\times x+P(\text{non - blue})\times(-2)=0$.

Step3: Substitute values and solve

Substitute $P(\text{blue})=\frac{1}{5}$ and $P(\text{non - blue})=\frac{4}{5}$ into the equation: $\frac{1}{5}x+\frac{4}{5}\times(-2)=0$. First, simplify the equation: $\frac{1}{5}x-\frac{8}{5}=0$. Then add $\frac{8}{5}$ to both sides: $\frac{1}{5}x=\frac{8}{5}$. Multiply both sides by 5 to get $x = 8$.

Answer:

C. 8