using a fair coin). But there are many conceivable combinations of totals that would give that average. Half the trials could turn up 10 heads, for instance, while the other half turned up zero every time. Or you could imagine getting precisely 5 heads in every 10-flip trial.
What actually happens is that the number of trials with different numbers of heads is distributed all across the board, but with differing probabilities—about 25 percent of the time you’ll get 5 heads, 20 percent of the time 4 (same for 6), 12 percent of the time 3 (also for 7). You would expect to get 1 head 1 percent of the time (and no heads at all out of a 10-flip run about 0.1 percent of the time, or once in a thousand). Coin tossing, in other words, produces a probability distribution of outcomes, not merely some average outcome. Maxwell’s insight was that the same kind of probability distribution governs the possible allocations of energy among a mess of molecules. And game theory’s triumph was in showing that a probability distribution of pure strategies—a mixed strategy—is usually the way to maximize your payoff (or minimize your losses) when your opponents are playing wisely (which means they, too, are using mixed strategies).
Imagine you are repeatedly playing a simple game like matching pennies, in which you guess whether your opponent’s penny shows heads or tails. Your best mixed strategy is to choose heads half the time (and tails half the time), but it’s not good enough just to average out at 50-50. Your choices need to be made randomly, so that they will reflect the proper probability distribution for equally likely alternatives. If you merely alternate the choice of heads or tails, your opponent will soon see a pattern and exploit it; your 50-50 split of the two choices does you no good. If you are choosing with true randomness, 1 percent of the time you’ll choose heads 9 times out of 10, for instance.
In his book on behavioral game theory, Colin Camerer discusses studies of this principle in a real game—tennis—where a similar 50-50 choice arises: whether to serve to your opponent’s right or left side. To keep your opponent guessing, you should serve one way or the other at random.22 Amateur players tend to