Final answer:
The probability that a student is on the green team twice is determined by multiplying the probability of a student being on the green team in the first game by the probability of being on the green team in the second game. These probabilities would be the same if marbles are replaced after each draw, or different if they are not replaced.
Step-by-step explanation:
The student's question involves finding the probability of a specific outcome in a random experiment, which is a fundamental concept in mathematics, particularly in the branch known as probability theory. Since each student's choice is independent of the other, and assuming that the number of green marbles remains constant for each game, the probability of a student being on the green team for one game is the number of green marbles divided by the total number of marbles. If the probabilities are the same for each game, the overall probability that a student is on the green team twice is the product of the two individual probabilities.
For instance, if there are 4 green marbles and the total marbles are 7 (4 green + 3 white), the probability of choosing a green marble once would be 4/7. To be on the green team twice, the student must get green both times, so we multiply the single occurrence probability by itself: (4/7) * (4/7), which gives the probability of being on the green team twice.
If marbles are removed after being chosen (as in the examples provided), the total number of marbles and their respective colors would change for the second choice, thus affecting the probability of the second draw. This scenario is different from replacement draws, where the probabilities remain the same for each draw because the composition of the bag does not change.