Question

mia place two green marbles and one yellow marble in a bag. derek picks a marble from 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
p(gg) = 4/9
complete
p(gy) = 2/9
complete
p(yg) =
done

Explanation:

Step1: Calculate probability of first - pick yellow and second - pick green

There are 2 green marbles and 1 yellow marble, so total 3 marbles.
The probability of picking a yellow marble on the first pick is $P(Y_1)=\frac{1}{3}$ since there is 1 yellow marble out of 3 marbles.
Since the marble is replaced, the probability of picking a green marble on the second pick is $P(G_2)=\frac{2}{3}$ as there are 2 green marbles out of 3 marbles.

Step2: Use multiplication rule for independent events

For independent events, the probability of event A and event B occurring is $P(A\cap B)=P(A)\times P(B)$. Here, $P(YG) = P(Y_1)\times P(G_2)$.
$P(YG)=\frac{1}{3}\times\frac{2}{3}=\frac{2}{9}$

Answer:

$\frac{2}{9}$