One divided by infinity is not zero? I know that $\frac{1}{\infty}$ is undefined. But my question is - can we say that $\frac{1}{\infty}\neq0$ ?

I've got some idea how to explain that: Let's say we have a random-number generator that generates numbers in interval $(0, \infty)$. What is the probability that it generates 5? $\frac{1}{\infty}$. So it cannot equals zero because if it was zero, it wouldn't generate any number (every number has the same probability - $\frac{1}{\infty}$).

Am I right? Or is that wrong idea?

Thanks
It is worth noting that since $\infty$ is not a number, the expression $\frac{1}{\infty}$ is not meaningful. That is, you cannot evaluate this expression. It is not a number. You can think of it in much the same way as $\frac{1}{0}$ --- an expression without any interpretation.

The only means that we have to talk about expressions involving $\infty$ is through the concept of limits. The shorthand
$$ \lim_{x \to \infty} \frac{1}{x} = \frac{1}{\infty} = 0 $$
is sometimes used in a careless way by students, but the middle expression in this string of equations isn't strictly meaningful. It is only a device used to remind you that we are looking at what happens as we divide 1 by increasingly large numbers.

So, I suppose that in some sense, you can say that $\frac{1}{\infty} \neq 0$, as the left-hand side here isn't any number at all.
If we are to understand "$=$" as a binary relation of identity and "$\neq$ as a binary relation of non-identity, then in most contexts, formally, $1/\infty$ is not identical to zero. Therefore it would make logical sense to write $1/\infty \neq 0$ as long as we "define what we mean by $1/\infty$". This is more in a strictly logical sense than in a mathematical sense.

In order to interpret $1/\infty \neq 0$ mathematically, you have to carefully describe the underlying structure of the statement.

If we are to only consider "$=$" and "$\neq$" for numbers or algebraic expressions involving numbers, then $1/\infty \neq 0$ doesn't make sense because the left side isn't a number. However, if you define $1/\infty$ appropriately (e.g. as a limiting procedure), then the statement makes sense and can be evaluated as false.