derek and mia place two green marbles and one yellow marble in a bag. somebody picks a marble out of the bag…

Explanation:

Step1: Calculate total number of outcomes

There are 3 marbles, and since the marble is replaced after each pick, for two - pick events, the total number of outcomes is $3\times3 = 9$ according to the multiplication principle.

Step2: Calculate number of same - color outcomes

For green - green: The probability of picking a green on the first pick is $\frac{2}{3}$, and since the marble is replaced, the probability of picking a green on the second pick is also $\frac{2}{3}$. So the probability of green - green is $\frac{2}{3}\times\frac{2}{3}=\frac{4}{9}$. For yellow - yellow: The probability of picking a yellow on the first pick is $\frac{1}{3}$, and the probability of picking a yellow on the second pick is $\frac{1}{3}$. So the probability of yellow - yellow is $\frac{1}{3}\times\frac{1}{3}=\frac{1}{9}$. The probability of getting the same color is $\frac{4 + 1}{9}=\frac{5}{9}$.

Step3: Calculate number of different - color outcomes

The probability of getting different colors is $1-\frac{5}{9}=\frac{4}{9}$.

Step4: Define the value of points

Let $X$ be the random variable representing Derek's points. When the colors are the same, Derek gets $- 1$ point (probability $p_1=\frac{5}{9}$), and when the colors are different, Derek gets $1$ point (probability $p_2=\frac{4}{9}$).

Step5: Calculate the expected value

The formula for the expected value $E(X)$ of a discrete random variable is $E(X)=\sum_{i}x_ip_i$. Here, $E(X)=(-1)\times\frac{5}{9}+1\times\frac{4}{9}=\frac{-5 + 4}{9}=-\frac{1}{9}$.

Answer:

$-\frac{1}{9}$