What is the probability of a four occurring in 300 dice rolls?

Ask the complement question:

What is the probability that a 4 will not occur?

That, of course, is $\left(\frac56\right)^{300}$. So the probability of rolling at least one 4 is $$1-\left(\frac56\right)^{300}=1-1.76046×10^{-24}$$ It really is so close to 1 that I had to resort to just writing the difference out – the raw probability cannot be distinguished from 1 in 64-bit floating point.

Hint: What is the probability of NOT rolling a single 4?

The easiest way to look at this is first to compute the probability that none of the outcomes is a $4$. So for each roll there would then be five possibilities out of six and for two goes this would be $\left(\frac 56\right)^2$ and the probability of at least one $4$ would be $1-\left(\frac 56\right)^2= 1 - \frac {25}{36} = \frac {11}{36}$.

Now apply this thinking to your main case.