Question

finding the expected value in a marble game
derek and mia place two green marbles and one yellow marble in a bag. somebody picks a marble out of the bag without looking and records its color (g for green and y for yellow). they replace the marble and then pick another marble. if the two marbles picked have the same color, derek loses 1 point and mia gains 1 point. if they are different colors, mia loses 1 point and derek gains 1 point. what

Explanation:

Step1: Calculate probability of GG

There are 2 green and 1 yellow marbles, so total 3 marbles. Probability of picking green first is $\frac{2}{3}$, and since marble is replaced, probability of picking green second is also $\frac{2}{3}$. So $P(GG)=\frac{2}{3}\times\frac{2}{3}=\frac{4}{9}$.

Step2: Calculate probability of GY

Probability of picking green first is $\frac{2}{3}$, and probability of picking yellow second is $\frac{1}{3}$. So $P(GY)=\frac{2}{3}\times\frac{1}{3}=\frac{2}{9}$.

Step3: Calculate probability of YG

Probability of picking yellow first is $\frac{1}{3}$, and probability of picking green second is $\frac{2}{3}$. So $P(YG)=\frac{1}{3}\times\frac{2}{3}=\frac{2}{9}$.

Step4: Calculate probability of YY

Probability of picking yellow first is $\frac{1}{3}$, and since marble is replaced, probability of picking yellow second is also $\frac{1}{3}$. So $P(YY)=\frac{1}{3}\times\frac{1}{3}=\frac{1}{9}$.

Step5: Calculate expected - value

Let $X$ be Derek's score. If the colors are the same (GG or YY), Derek loses 1 point, if different (GY or YG), Derek gains 1 point.
$E(X)=(- 1)\times(P(GG)+P(YY))+1\times(P(GY)+P(YG))$.
Substitute the values: $E(X)=(-1)\times(\frac{4}{9}+\frac{1}{9}) + 1\times(\frac{2}{9}+\frac{2}{9})$.
$E(X)=(-1)\times\frac{5}{9}+1\times\frac{4}{9}=-\frac{5}{9}+\frac{4}{9}=-\frac{1}{9}$.

Answer:

$-\frac{1}{9}$