How many ideas hover dispersed in my head of which many a pair, if they should come together, could bring about the greatest of discoveries! But they lie as far apart as Goslar sulphur from East India saltpeter, and both from the dust in the charcoal piles on the Eichsfeld — which three together would make gunpowder. How long the ingredients of gunpowder existed before gunpowder did! There is no natural aqua regia. If, when thinking, we yield too freely to the natural combinations of the forms of understanding and of reason, then our concepts often stick so much to others that they can’t unite with those to which they really belong. If only there were something in that realm like a solution in chemistry, where the individual parts float about, lightly suspended, and thus can follow any current. But since this isn’t possible, we must deliberately bring things into contact with each other. We must experiment with ideas.

— G.C. Lichtenberg, Aphorisms

A drunk man arrives at his doorstep and tries to unlock his door. There are 10 keys on his key ring, one of which will fit the lock. Being drunk, he doesn’t approach the problem systematically; if a given key fails to work, he returns it to the ring and then draws again from all 10 possibilities. He tries this over and over until he gets the door open. Which try is most likely to open the door?

Surprisingly, the first try is most likely. The probability of choosing the right key on the first try is 1/10. Succeeding in exactly two trials requires being wrong on the first trial and right on the second, which is less likely: 9/10 × 1/10. And succeeding in exactly three trials is even less likely, for the same reason. The probability diminishes with each trial.

“In other words, it is most likely that he will get the right key at the very first attempt, even if he is drunk,” writes Mark Chang in Paradoxology of Scientific Inference. “What a surprise!”