Final answer:
To solve this problem, we can create a tree diagram to represent all the possible outcomes for drawing two marbles without replacement. From the tree diagram, we can calculate the probabilities for each scenario: P(B lue, Blue) = 5/22, P(Blue, White) = 2/11, P(White, Green) = 2/33. The probability of drawing the same color twice is 1/6, and the probability that the two marbles of the same color came from White is 4/11.
Step-by-step explanation:
To solve this problem, we can create a tree diagram to represent all the possible outcomes for drawing two marbles without replacement.
a) To find the probability of drawing two blue marbles, we follow the path on the tree diagram that represents drawing a blue marble on the first and second draw. There are 6 blue marbles initially and 5 after the first draw, so the probability is (6/12) * (5/11) = 30/132 = 5/22.
b) To find the probability of drawing a blue marble on the first draw and a white marble on the second draw, we follow the path on the tree diagram that represents this sequence. There are 6 blue marbles initially and 4 white marbles initially, so the probability is (6/12) * (4/11) = 24/132 = 2/11.
c) To find the probability of drawing a white marble on the first draw and a green marble on the second draw, we follow the path on the tree diagram that represents this sequence. There are 4 white marbles initially and 2 green marbles initially, so the probability is (4/12) * (2/11) = 8/132 = 2/33.
d) To find the probability of drawing the same color twice, we need to sum up the probabilities of drawing two blue marbles, two white marbles, and two green marbles. From the tree diagram, we can see that the probability of drawing two blue marbles is 5/22, the probability of drawing two white marbles is 6/33, and the probability of drawing two green marbles is 1/66. Therefore, the probability of drawing the same color twice is (5/22) + (6/33) + (1/66) = 1/6.
e) To find the probability that the two marbles of the same color came from white, we consider the probability of drawing two white marbles and divide it by the probability of drawing the same color twice. From the tree diagram, we can see that the probability of drawing two white marbles is 6/33, and the probability of drawing the same color twice is 1/6. Therefore, the probability that the two marbles of the same color came from white is (6/33) / (1/6) = 12/33 = 4/11.