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What is the probability to get $n$ times the same element $k$ choosing randomly from a set $A$ knowing the cardinality $|A| = c$?

Thank you in advance,
rubik

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2 Answers 2

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It depends: is $k$ fixed in advance? if so then the probability is given by $\frac{1}{c^n}$. If you just ask for the probability to have $n$ times the same element, then it is $\frac{1}{c^{n-1}}$ (since the first time you can have any object). By the way, I supposed you are doing exactly $n$ tries. Of course, the probability changes if you are doing more than $n$ tries.

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  • $\begingroup$ Thank you! But why does it change if $k$ is fixed? $\endgroup$
    – rubik
    Nov 2, 2010 at 15:33
  • $\begingroup$ @rubik: if $k$ is fixed IN ADVANCE, then the first time you must choose it. If $k$ is not decided in advance, then you just ask that starting from the second draw you get the same object as the first time. Thus, the first draw can be anyting. $\endgroup$
    – Djaian
    Nov 2, 2010 at 15:38
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Assuming that you draw $n$ times uniformly at random from the set of $c$ elements, it is $\frac{1}{c^{n-1}}$ as Djaian says. You are rolling a $c$-sided die.

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  • $\begingroup$ Thank you! I have accepted Djaian's answer because he answered first. Thank you anyway. $\endgroup$
    – rubik
    Nov 2, 2010 at 15:32

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