TITLE: Chances of rolling "Snake Eyes" at least once in a series of rolls.
QUESTION [10 upvotes]: So I know that if you roll a standard pair of dice, your chances of getting Snake Eyes (double 1s) is $1$ in $36$. What I'm not sure of is how to do the math to figure out your chances of rolling Snake Eyes at least once during a series of rolls. I know if I roll the dice $36$ times it won't lead to a $100\%$ chance of rolling Snake Eyes, and while I imagine it's in the upper nineties, I'd like to figure out exactly how unlikely it is.

REPLY [32 votes]: The other answers explain the general formula for the probability of never rolling snake eyes in a series of $n$ rolls.
However, you also ask specifically about the case $n=36$, i.e. if you have a $1$ in $k$ chance of success, what is your chance of getting at least one success in $k$ trials? It turns out that the answer to this question is quite similar for any reasonably large value of $k$. 
It is $1-\big(1-\frac{1}{k}\big)^k$, and $\big(1-\frac{1}{k}\big)^k$ converges to $e^{-1}$. So the probability will be about $1-e^{-1}\approx 63.2\%$, and this approximation will get better the larger $k$ is. (For $k=36$ the real answer is $63.7\%$.)