  Bell's theorem supposedly demonstrates an irreconcilable conflict between
quantum mechanics and local, realistic hidden variable theories. In this paper
we show that all experiments that aim to prove Bell's theorem do not actually
achieve this goal. Our conclusions are based on a straightforward statistical
analysis of the outcomes of these experiments. The key tool in our study is
probability theory and, in particular, the concept of sample space for the
dichotomic random variables that quantifies the outcomes of such experiments.
We also show that an experimental proof of Bell's theorem is not, in principle,
impossible, but it would require a completely different experimental apparatus
than those commonly used to allegedly achieve this objective. The main
consequence of our work is that we cannot dismiss local realistic hidden
variable theories on the basis of currently available experimental data.
