# Black and white balls in the urn. What is the probability of…?

There are $100$ balls in the urn - $50$ black and $50$ white. A person randomly chooses a batch of $50$ balls and replaces it with a batch of inversely colored balls (i.e. for example a batch of $34$ black and $16$ white balls would be replaced by a batch of $16$ black and $34$ white balls).

It is obvious that after replacement took place there could be anything from all black to all white balls. But what is the probability that one can still find at least $40$ balls of either color?

Honestly I'm not even sure how to figure out the probability of urn still having 50-50 balls, and for my original question apparently I have to sum the probabilities of $60-40$, $59-41$, $58-42$, $\ldots$, $41-59$, $40-60$ ball combinations.

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If the 50 chosen include $w$ white, then they include $50-w$ black, and the remaining 50 are $50-w$ white and $w$ black, so after the replacement you have $100-2w$ white and $2w$ black. So you need to compute, for all values of $w$ that makes both $100-2w$ and $2w$ at least 40, the probability that 50 drawn at random have $w$ white. Then add. –  Gerry Myerson Jul 9 '13 at 9:03
I can with certainty say that the probability of 59-41, 57-43, 55-45, 53-47 , ..., 41-59 occurrence are all zero! –  Ali Jul 9 '13 at 9:06
$\frac{35973547957477625884325468}{36983630698520598729769977} \simeq 0.972688$ –  Ali Jul 9 '13 at 9:22

If the person chooses $k$ black and $50-k$ white balls, then after inverted replacement, we have $100-2k$ black and $2k$ white balls.
You will find at least $40$ balls, iff $2k\geq 40$ or $100-2k\geq 40$, in other words, iff $20\leq k\leq 30$.
There are $\binom{100}{50}$ possible batches. Now count the batches with exactly $k$ black balls. You can choose the $k$ black balls in $\binom{50}{k}$ ways and the $50-k$ white balls in $\binom{50}{50-k}=\binom{50}{k}$, ways. So the result should be: $$\sum\limits_{k=20}^{30}\frac{\binom{50}{k}^2}{\binom{100}{50}}\approx 97\%$$
A little slip: you need $20\le k \le 30$ –  Michael Jul 9 '13 at 9:18
@GerryMyerson Not sure about that. If the batch consisted of only $10$ balls then apparently $\sum\limits_{k=0}^{10}\frac{\binom{50}{k}^2}{\binom{100}{10}}>1$ . What is the problem? –  Pranasas Jul 10 '13 at 13:03