For some purposes, the [weak law of large numbers][1] is superior to the strong law. This is one of those purposes. The weak law is quantitative, unlike the strong law.
Let $X$ be the arithmetic mean of a random sample of size $n$ chosen from a given distribution with mean $\mu$ and variance $\sigma^2$. Then for any $\epsilon>0$ $$ P(|X-\mu| \ge \epsilon) \le \frac{\sigma^2}{ n \epsilon^2} $$
Example: tossing a fair coin (mean $\mu=1/2$, variance $\sigma=1/4$), I want to get within $\epsilon = 0.1$ of the mean $1/2$. If I toss it $n=1000$ times, then I will be within that error with probability at least $$ 1-\frac{\sigma^2}{n\epsilon^2} = \frac{159}{160} \approx 0.99 $$
Now $X = H/n$ where $H$ is the number of times my coin came up on heads. So this means (fair coin, tossing 1000 times) $$ P(400 < H < 600) > 0.99 $$ [1]: http://mathworld.wolfram.com/WeakLawofLargeNumbers.html