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A box contains $20$ balls all of different colors including the red color. If we select $10$ balls randomly without replacement, what is the probability that the red ball will be among these $10$ balls?

What I think is that: If we let $X$ to be the number of balls we select until we get the red ball, then $X$ will be a random variable with range $ 1,2,3, \ldots ,20 $, and the probability of getting the red ball will be $1/20$, so our probability will be $$\left(\frac{1}{20} \right)^{10} ,$$ is that right?! I'm not sure about the distribution of $X$?

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    Your argument is wrong. But how do we check that? Ok, instead of picking just 10 balls, suppose I pick all the 20 balls. Without any calculations, what do you think is the probability the red ball is among the 20 balls? Now, what does your argument tell you?2011-09-19
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    If you select all the 20 balls then the probability is 1. So my argument is false!2011-09-19
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    Not just that. As you pick more balls, the probability should clearly increase. Your answer $$\left(\frac{1}{20} \right)^{\text{number of balls picked}}$$ is *decreasing* as I pick more balls and so it must be wrong. (If you change the answer to $1$ minus that value, then it is still wrong, but it is slightly better.)2011-09-19

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