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For every integer $n>1$, I have a die $D_n$ with $a_n$ sides labeled $1$ to $a_n$. If $a_n=n^k$, for integer $k>0$, and I roll all the dice at once, what is the probability none of them lands on the side labeled $1$? What is the probability exactly one lands on the side labeled $1$?

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    These are not infinite dice. Each one is finite. I think you meant _infinitely many dice_. It seems to be a widespread error to say "infinite Xs" when what is meant is "infinitely many Xs". Widespread, but still an error.2011-11-10
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    I see that's been corrected now.2011-11-10
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    These dice are independent? Do you know a formula for events (from a sequence of independent events) all happening? As this is a homework-type problem, I give these hints and not a full solution.2011-11-10

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