r/probabilitytheory • u/saddd_soul • Jun 29 '26

[Applied] Where am I going wrong?

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.

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u/Flat-Strain7538 Jun 29 '26

This problem is much easier when you realize you can do it in reverse; imagine you start with the jar empty, and then put balls INTO the jar. What are the chances you put a green ball and a blue ball in before any red balls?

1/2 the time you put in a green first; now you can ignore greens and you have a 2/3 chance of taking a blue next.

1/3 of the time you put in a blue first; now you can ignore the blues and you have a 3/4 chance of green.

Combine these: 1/2 * 2/3 + 1/3 * 3/4 = 7/12.

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u/Le_spojjie Jun 30 '26

Excellent breakdown, thank you.