Chris at Smart Football has discussed the merits of calling plays randomly on a couple occasions, including earlier this month where he found some support for it from Bill Walsh. Essentially, the argument in favor of it breaks down like this: in a game where everyone is trying to out-think one another, the only way to eliminate the advantage of superior thinkers is to do things randomly.
I've been vaguely in favor of this, but I saw something this morning on Slashdot that really helped the principle hit home. It's the Tuesday Birthday Problem, and it has generated a lot of heated discussion among mathematicians. Simply put, it is this:
I have two children, one of whom is a son born on a Tuesday. What is the probability that I have two boys?
The classic answer to this problem is 1/3. There are four equally likely possibilities for the birth order of any two children: boy-boy, boy-girl, girl-boy, and girl-girl. The last one is impossible because one child is a boy, so three possibilities remain. Among those only one has a boy-boy outcome, therefore the probability is 1/3.
A Stanford mathematician went all-out and took the day of the week into account. He used the assumption that by specifying the day of the week the one child was born on, the other must not have been born on a Tuesday. In that scenario, the probability of two boys is 13/27, which is almost 1/2. Looking at it that way, it suggests that the classic answer to the puzzle is wrong.
There's a good reason for that: it is wrong*. Assuming that the there's an equally likely chance of a boy or girl birth (and that the two kids aren't twins), the probability of the second child being a boy is 1/2. The probability breakdown of the three options listed in the classic answer really isn't equal; it's 50% for boy-boy and 50% for not boy-boy, with a 25% chance each for the boy-girl and girl-boy orders. The gender of the unknown child is completely independent of your knowledge of the other child's gender, and it conforms to the 50-50 chances of all children.
So what does this have to do with play calling? I think this problem, along with Car Talk's Monty Hall problem, shows that our minds are not wired for dealing with true randomness. In fact, sometimes an answer that feels elegant (like the classic solution above) can be flat-out wrong debatable. It takes study and discipline to deal with it properly, and in the heat and high emotional level of a game, it's likely that opposing coaches and especially players will not have that discipline at all times.
They'll likely even try to read into things and find patterns that aren't there, like a person fully expecting to see tails after flipping a coin and getting heads ten times in a row. The coin isn't due for tails; the probability for tails on every flip is still 50%. People are hard wired for finding patterns even when they're not there, and it's difficult to overcome that.
As Chris mentions, doing things truly randomly is probably not a good idea. He points out that you'd want to pick randomly from plays that complement and counter each other. I'd also add that true randomness might get you calling a Hail Mary play on first-and-goal from the two yard line. You might catch the defense off guard with that, but your likelihood of scoring would be much less than if you called the utterly predictable halfback dive.
Ultimately I doubt we'll ever see anyone call plays with true randomness, as you'd need a Mike Leach-sized playbook or an even smaller one to make it manageable. Perhaps just flipping a coin for running and passing could work, but as someone who's never coached, I can't tell you how well that would work out.
If it ever does happen, it probably would occur in the one of the vast laboratories known as high school football and small college football. Many of the innovations in offense of the past couple decades have come from those locales. I think it's worth a shot; you literally can't know for sure what will happen.
*Note: This is a riddle with a lot of different right answers. The one I posted is the one I think is most correct (because I view the children's births not as a set of probabilistic events but as two independent events), but it is by no means the only answer. I should not have made it sound like it was the only one, because it's not.