[ig06]

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I think you didn't really bother to read the question which was chances of taking both balls from the bag with ten red balls.

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trickster, that isn't correct.I think you should be careful when making such statements as the above...
BruceZ's answer is correct. You are confusing two questions. The question is NOT: "What is the probability of picking a ball from the bag with all red balls both times?". The question is: "GIVEN that I have picked 2 red balls, what is the probability that I picked from the bag with all red balls both times?"
These are absolutely not the same thing but its a mistake that many people who don't use Bayesian statistics on a regular basis often make.
You are working out P(all red bag both times)=1/4
The question asks for P(all red bag both times|both balls picked were red) where P(A|B) means probability that A is true given that B is true.
This is computed using P(A|B)=P(B|A)P(A)/P(B) which is what BruceZ did without explicitly stating it (I actually thought it was a very clear explanation).
If this isn't clear (and if you care!), google "Bayes Theorem" and think about it for a while.