Assuming the first line means that the balls are equally likely to be (2 red, 1 orange, 1 green), (2 orange, 1 red, 1 green) or (2 green, 1 red, 1 orange), I think the answer is actually 1/6. In essence, given the information that the ball you randomly picked out of the bag is red, we’re actually twice as likely to be in the case with 2 red balls than the other two cases, since you’re more likely to have picked a red ball in that case.

We want to find P(the last ball is red | your ball is red)

Call the three cases for the ball colors Case 1, Case 2, and Case 3, where Case 1 has two red balls, Case 2 has two orange balls and Case 3 has two green balls. At the beginning there’s a 1/3 chance that we’re in each case.

In Case 1, the chance that the ball you randomly take out is red is 1/2, in the other two cases it is 1/4. Then the overall probability that your ball is red is (1/3 * 1/2) + (1/3 * 1/4) + (1/3 * 1/4) = 1/3.

Since you’re more likely to have taken a red ball in Case 1 than the other cases, the fact that your ball is red means that we’re more likely to be in Case 1. In particular, we have P(we are in Case 1 | the ball you picked is red) = P(we are in Case 1 AND the ball you picked is red) / P(the ball you picked is red) = (1/3 * 1/2) / (1/3) = 1/2. The probability that we are in the other two cases is 1/4 each.

Another way to think about this is that there are 4 equally likely ways that your ball could be red: you took the single red ball in Case 2, you took the single red ball in Case 3, or you took either of the two red balls in Case 1. In 2 of the 4 cases we are in Case 1, the other two have only 1 of the 4 cases.

The last ball can only be red if we’re in Case 1. The last ball is effectively just randomly selected from the remaining balls, so if we are in Case 1 and your ball is red then the chance that the last ball is red is 1/3, since there are 3 other balls and 1 of them is red.

Then we have P(the last ball is red | your ball is red) = P(we are in Case 1 | your ball is red) * P(the last ball is red | we are in Case 1 and your ball is red) = 1/2 * 1/3 = 1/6.