To find the expected value, we need to multiply the value of each possible outcome by its probability, and then sum these products. Let's first calculate the probability of selecting each type of marble:
P(gold) = 3 / (3 + 7 + 20) = 0.097
P(silver) = 7 / (3 + 7 + 20) = 0.226
P(black) = 20 / (3 + 7 + 20) = 0.676
Now we can calculate the expected value:
E(X) = $3 * P(gold) + $2 * P(silver) - $1 * P(black)
= $3 * 0.097 + $2 * 0.226 - $1 * 0,676
= $0.291 + $0.452 - $0676
= $0.067
Therefore, the expected value of playing this game is $0.067. This means that on average, for every game played, you would expect to win about 6.7 cents. Since the expected value is positive, it is a favorable game for you to play, but keep in mind that this does not guarantee that you will win money in any given play.