Consider that 1) There is more than one possible arrangement of proteins which qualifies as a living cell, and that 2) the materials of which proteins are made had quite a long time to shuffle around and try out different configurations between when the earth cooled and the present day, to say nothing of other planets elsewhere in the universe, and that 3) once a living, self-replicating, self-repairing cell has come to exist in an area with appropriate raw materials and a steady energy source it will create more such cells, so it only has to happen once.

So, we're looking at a sample size equal to, by my back-of-the-envelope estimation, the number of cell-sized volumes in Earth's atmosphere and oceans, times the number of planck instants in a little over four billion years, times the number of earth-like planets in the universe. The actual universe, not just the part we can see.

For intelligent design to be the most reasonable explanation, the probability of life emerging spontaneously would have to be low enough that, in a sample of that size, we wouldn't expect to see it happen even once, and, furthermore, the designer's own origin would need to be explained in such a way as to be less improbable.