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If the first blind pick is a purple ball then this has a probability of 6/10. If the second blind pick is purple ball then this has a probability of 5/9. If the third blind pick is another purple ball then this has a probability of 4/8. And the probability of 3/7 is the case of drawing fourth purple ball. The probability of the four purple balls being drawn is then 6/10*5/9*4/8*3/7 = 0.0714... I think that's the answer.
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