# Probability picking colored balls and removing all balls of the same color

I have $120$ colored balls. $5$ are $white$, $10$ are $red$, $15$ are $blue$, $15$ are $yellow$, $20$ are $green$, $25$ are $purple$ and $30$ are $black$.

I choose randomly a ball from those, and after doing that I remove all balls of that color.

What's the probability of picking a white ball after five picks?

It's clear that for only one pick, it's ${5 \over120}$. I managed to calculate it also for the second pick, which would be: $${5 \over 120 }+{115 \over 120} ({10\over120}{5 \over110}+2{15\over120}{5 \over105}+{20\over120}{5 \over100}+{25\over120}{5 \over95}+{30\over120}{5 \over90})$$

But can't go further.

• It's been more than a year since the question and nobody seems to care about this question. I'd like to know if it's because it's too confusing, too tedious to solve or because it's simply not interesting. That way I'd be able to ask a better question. – Masclins Jan 22 '18 at 15:24
• I'd suggest simplifying down to, say, four colors and two picks, for which you could do it with 24 branches (4 colors first pick, 3 for each of those, etc). See if there is an identifiable pattern. If there isn't some nice pattern, and it's just a big tree calculation, then it isn't probably interesting to do it for larger numbers of colors. – Ned Jan 22 '18 at 22:05