# Continuous Dice rolls

A dice is being rolled continuously. Suppose you roll a 3 ( or any number ). What is the probability that you will get the next "3" exactly after n rolls?

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I suspect you mean "A die is being rolled continually". –  Robert Israel Jul 28 '12 at 1:48
You are right.. I do mean continually. –  Chip Jul 28 '12 at 7:48

I will interpret "after $n$ rolls" as meaning that we want the probability that our next $n-1$ rolls do not yield a $3$, but the $n$-th roll does give a $3$..
The probability of a non-$3$ on any toss is $\frac{5}{6}$. So the probability of $n-1$ consecutive non-$3$'s followed by a $3$ is $$\left(\frac{5}{6}\right)^{n-1}\frac{1}{6}.$$
We can also get the result by using a counting argument. Imagine tossing a die $n$ times, and recording the results. So a record of $4$ tosses might read $4,5,4,1$.
There are $6^n$ possible records of the results of $n$ tosses, all equally likely. There are $5^{n-1}$ records in which the first $n-1$ results are a non-$3$ and the last result is a $3$. So our probability is $$\frac{5^{n-1}}{6^n}.$$