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?\no 4\no 6\no 8\no 10

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?\no 4\no 6\no 8\no 10

Answer

Explanation:

Step1: Calculate probability of losing

There are 4 non - blue marbles out of 5 marbles. Probability of losing $P(L)=\frac{4}{5}$.

Step2: Calculate probability of winning

There is 1 blue marble out of 5 marbles. Probability of winning $P(W)=\frac{1}{5}$.

Step3: Set up expected - value equation for a fair game

Let the value of the blue marble be $x$ points. For a fair game, the expected value $E = 0$. The value of losing is - 2 points. So, $E=P(W)\times x+P(L)\times(- 2)=0$. Substitute the probabilities: $\frac{1}{5}x+\frac{4}{5}\times(-2)=0$.

Step4: Solve the equation for $x$

First, expand the equation: $\frac{1}{5}x-\frac{8}{5}=0$. Add $\frac{8}{5}$ to both sides: $\frac{1}{5}x=\frac{8}{5}$. Multiply both sides by 5 to solve for $x$: $x = 8$.

Answer:

C. 8