Answer:
Thus, the expected value of points for Derek and Mia are
and
respectively.
Step-by-step explanation:
Number of green marbles = 2 and Number of Yellow marbles = 1
Then, total number of marbles = 2+1 = 3
A person selects two marbles one after another after replacing them.
So, the probabilities of selecting different combinations of colors are,
![1.\ P(GG)=P(G)\times P(G)\\\\P(GG)=\dfrac{2}{3}\times \dfrac{2}{3}\\\\P(GG)=\dfrac{4}{9}](https://tex.z-dn.net/?f=1.%5C%20P%28GG%29%3DP%28G%29%5Ctimes%20P%28G%29%5C%5C%5C%5CP%28GG%29%3D%5Cdfrac%7B2%7D%7B3%7D%5Ctimes%20%5Cdfrac%7B2%7D%7B3%7D%5C%5C%5C%5CP%28GG%29%3D%5Cdfrac%7B4%7D%7B9%7D)
![2.\ P(GY)=P(G)\times P(Y)\\\\P(GY)=\dfrac{2}{3}\times \dfrac{1}{3}\\\\P(GY)=\dfrac{2}{9}](https://tex.z-dn.net/?f=2.%5C%20P%28GY%29%3DP%28G%29%5Ctimes%20P%28Y%29%5C%5C%5C%5CP%28GY%29%3D%5Cdfrac%7B2%7D%7B3%7D%5Ctimes%20%5Cdfrac%7B1%7D%7B3%7D%5C%5C%5C%5CP%28GY%29%3D%5Cdfrac%7B2%7D%7B9%7D)
![3.\ P(YG)=P(Y)\times P(G)\\\\P(YG)=\dfrac{1}{3}\times \dfrac{2}{3}\\\\P(YG)=\dfrac{2}{9}](https://tex.z-dn.net/?f=3.%5C%20P%28YG%29%3DP%28Y%29%5Ctimes%20P%28G%29%5C%5C%5C%5CP%28YG%29%3D%5Cdfrac%7B1%7D%7B3%7D%5Ctimes%20%5Cdfrac%7B2%7D%7B3%7D%5C%5C%5C%5CP%28YG%29%3D%5Cdfrac%7B2%7D%7B9%7D)
![4.\ P(YY)=P(Y)\times P(Y)\\\\P(YY)=\dfrac{1}{3}\times \dfrac{1}{3}\\\\P(YY)=\dfrac{1}{9}](https://tex.z-dn.net/?f=4.%5C%20P%28YY%29%3DP%28Y%29%5Ctimes%20P%28Y%29%5C%5C%5C%5CP%28YY%29%3D%5Cdfrac%7B1%7D%7B3%7D%5Ctimes%20%5Cdfrac%7B1%7D%7B3%7D%5C%5C%5C%5CP%28YY%29%3D%5Cdfrac%7B1%7D%7B9%7D)
Now, we have that,
If two marbles are of same color, then Mia gains 1 point and Derek loses 1 point.
If two marbles are of different color, then Derek gains 1 point and Mia loses 1 point.
<h3>Also, the expected value of a random variable X is
![E(X)=\sum_{i=1}^{n} x_i\times P(x_i)](https://tex.z-dn.net/?f=E%28X%29%3D%5Csum_%7Bi%3D1%7D%5E%7Bn%7D%20x_i%5Ctimes%20P%28x_i%29)
.</h3>
Then, the expected value of points for Derek is,
![E(D)= (-1)\times \dfrac{4}{9}+1\times \dfrac{2}{9}+1\times \dfrac{2}{9}+(-1)\times \dfrac{1}{9}\\\\E(D)= \dfrac{-5}{9}+\dfrac{4}{9}\\\\E(D)=\dfrac{-1}{9}](https://tex.z-dn.net/?f=E%28D%29%3D%20%28-1%29%5Ctimes%20%5Cdfrac%7B4%7D%7B9%7D%2B1%5Ctimes%20%5Cdfrac%7B2%7D%7B9%7D%2B1%5Ctimes%20%5Cdfrac%7B2%7D%7B9%7D%2B%28-1%29%5Ctimes%20%5Cdfrac%7B1%7D%7B9%7D%5C%5C%5C%5CE%28D%29%3D%20%5Cdfrac%7B-5%7D%7B9%7D%2B%5Cdfrac%7B4%7D%7B9%7D%5C%5C%5C%5CE%28D%29%3D%5Cdfrac%7B-1%7D%7B9%7D)
And the expected value of points for Mia is,
.
Thus, the expected value of points for Derek and Mia are
and
respectively.