These two expressions correspond to the weak and strong law of large numbers, respectively: $$ \lim_{n \rightarrow \infty} P(|\bar{X} - \mu| \geq \epsilon) = 0 \ \ \text{for } \epsilon > 0 \\ P(\lim_{n\rightarrow 1} \bar{X} = \mu) = 1 $$
I've always struggled to understand the difference between the two. It seems very subtle to me. The first one is a limit over the probability, where as the second one the limit is within the probability and it's a limit over $\bar{X}$, the sample mean for $n$ samples.
But I cannot wrap my head around how these two are saying slightly different things.
The first one is telling me that when the number of samples approach infinity, the probability that the sample mean is different from the population mean is zero.
The second one is telling me that the probability of the sample mean equating to the population mean is 1 when the number of samples approach infinity.
How are these two actually different? It reads to me pretty much the same.