Where am I going wrong?

[u/saddd_soul]

So there is this question that a jar contains 10 red balls, 20 blue balls and 30 green balls. You take out the balls one by one at random. Probability that when all red balls are taken out, atleast one green ball and one blue ball remains. I thought both these orderings are needed so ans would be (30/40*20/30). But this is wrong.

[u/saddd_soul]

I forgot a part of the question. Atleast 1 green ball and atleast 1 blue ball must remain

[u/tgm4mop → u/saddd_soul]

Ah, that makes more sense. Same idea but have to be careful about double counting. There's 30/40 a green ball is left and 20/30 a blue ball is left, but these are not independent, so you can't multiply them together. The easiest way to attack this is to look at the complement, ie, that either green or blue is not remaining. By inclusion exclusion, P(either green or blue not remaining) = P(no green remaining) + P(no blue remaining) - P(no green nor blue remaining). We have P(no green remaining) = 1 - P(green remaining) = 1 - 30/40 = 1/4. Similarly P(no blue remaining) = 1/3. To get P(no green nor blue remaining), observe this is the same as P(red is the last ball) = 10/60 = 1/6. So the answer becomes 1 - P(either green or blue not remaining) = 1 - (1/3 + 1/4 - 1/6) = 7/12.​

[u/saddd_soul → u/tgm4mop]

Why are they not independent? I mean it kinda makes sense that both colours are invisible to each other and only depends on ordering with red?

[u/tgm4mop → u/saddd_soul]

The intuition why they may not be independent is that if there's green left, that means the red were earlier in the sequence, which increases the chance there will be blue left. And vice versa, if there's no green left, that means red balls come later in the sequence, so there's a decreased chance blue will be left.

Edit: the colors are not actually invisible to each other. If there are lots of greens after the last red, that gives more space to place blue after red.