Step-by-step explanation:
a probability is always
desired cases / totally possible cases
4 red, 2 white, 3 green balls. that means we have
4+2+3 = 9 balls in the box.
the probability at the beginning to draw a red ball is
4/9
the probability at the beginning to draw a white ball is
2/9
the probability at the beginning to draw a green ball is
3/9 = 1/3
the same color actually means
2 red balls or 2 white balls or 2 green balls.
we need to look at these 3 different scenarios, and since they are completely separate events, we add them as the total probability.
the probability to get 2 red balls is
4/9 for the first pull.
then we have only 3 red balls out of now 8 total balls left, so the probability for the second pull is
3/8.
the combined probability for pulling 2 red balls is then
4/9 × 3/8 = 12/72 = 1/6
the probability to get 2 white balls is
2/9 for the first pull.
then we have 1 white ball out of now 8 total balls left.
the probability for the second pull is
1/8.
the combined probability for pulling 2 white balls is then
2/9 × 1/8 = 2/72 = 1/36
the probability to get 2 green balls is
3/9 for the first pull.
then we have 2 green balls out of now 8 total balls left.
the probability for the second pull is
2/8 (= 1/4).
the combined probability for pulling 2 green balls is then
3/9 × 2/8 = 6/72 = 1/12
now, the probability to get 2 marbles with the same color after 2 pulls is
12/72 + 2/72 + 6/72 = 20/72 = 5/18 = 0.277777777...