The strong and weak laws of large numbers: Why two?

Solution 1:

Weak laws are about convergence in probability and not convergence in distribution. Convergence in probability implies convergence in distribution but not the other way round.

One advantage of the weak law is that there need not be one probability space over which the entire sequence of random variables is simultaneously defined.

Thus suppose you want to formalise the statement “when tossing an unbiased coin about half of the tosses will be heads”. The weak law version can be proved in the context of elementary discrete probability by considering a sequence of probability space where a sample point in the $n$-th space consists of the outcome of the first $n$ tosses. Then by Chebyshev's inequality applied in this $n$-th space $$P(|S_n/n-1/2|>\epsilon)\le 1/(4n\epsilon^2)$$ and an elementary limit gives the weak law.

In the same setting the strong law would require us to consider infinite sequences of tosses as outcomes and that is not possible without using the tools of measure theory.

And sometimes it is not only technical convenience but the nature of the question itself which requires us to use different probability spaces for different values of $n$. see Example 6.3 in Billingsley's Probability and Measure.