Let’s try to be a bit clearer about what averaging means. Think of example is tossing a coin we discussed above in section 1.3.10.

To keep your attention, I’ll use the cheap ploy of involving money. Let’s play a game where if you get heads, you get $1 and if you get tails you pay me $1. On average, you’d expect not to make or lose any money. But how do we see this mathematically?

Well we assign a variable ${\colorbox[rgb]{1,1,1}{$x$}}_{\colorbox[rgb]{1,1,1}{$i$}}$ which is the amount of money you make or lose in the ith trial of a game. ${\colorbox[rgb]{1,1,1}{$x$}}_{\colorbox[rgb]{1,1,1}{$1$}}\colorbox[rgb]{1,1,1}{$,$}{\colorbox[rgb]{1,1,1}{$x$}}_{\colorbox[rgb]{1,1,1}{$2$}}\colorbox[rgb]{1,1,1}{$,$}\colorbox[rgb]{1,1,1}{$\mathrm{\dots}$}\colorbox[rgb]{1,1,1}{$,$}{\colorbox[rgb]{1,1,1}{$x$}}_{\colorbox[rgb]{1,1,1}{$N$}}$ each can take the values $\colorbox[rgb]{1,1,1}{$\pm $}\colorbox[rgb]{1,1,1}{$1$}$. You’re interested in the total amount you make or lose:

$$\colorbox[rgb]{1,1,1}{$X$}\colorbox[rgb]{1,1,1}{$=$}\colorbox[rgb]{1,1,1}{$\sum $}_{\colorbox[rgb]{1,1,1}{$i$}\colorbox[rgb]{1,1,1}{$=$}\colorbox[rgb]{1,1,1}{$1$}}^{\colorbox[rgb]{1,1,1}{$N$}}{\colorbox[rgb]{1,1,1}{$x$}}_{\colorbox[rgb]{1,1,1}{$i$}}$$ | (1.14) |

We’re going to average, $\colorbox[rgb]{1,1,1}{$\u27e8$}\colorbox[rgb]{1,1,1}{$\mathrm{\dots}$}\colorbox[rgb]{1,1,1}{$\u27e9$}$, this equation to get what you ”expect” to make.

$$\colorbox[rgb]{1,1,1}{$\u27e8$}\colorbox[rgb]{1,1,1}{$X$}\colorbox[rgb]{1,1,1}{$\u27e9$}\colorbox[rgb]{1,1,1}{$=$}\colorbox[rgb]{1,1,1}{$\u27e8$}\colorbox[rgb]{1,1,1}{$\sum $}_{\colorbox[rgb]{1,1,1}{$i$}\colorbox[rgb]{1,1,1}{$=$}\colorbox[rgb]{1,1,1}{$1$}}^{\colorbox[rgb]{1,1,1}{$N$}}{\colorbox[rgb]{1,1,1}{$x$}}_{\colorbox[rgb]{1,1,1}{$i$}}\colorbox[rgb]{1,1,1}{$\u27e9$}$$ | (1.15) |

That’s fine but we haven’t defined what we mean by average. That’s a bit tricky. It’s like the crossing-the-freeway example in subsection 1.2.1. You want to know how much on average you make or lose on the ith trial. There are a lot of ways to define the average, but what you’re interested in is what would happen if average over different times that you play the game. So that’s the kind of average that is most important to realists like ourselves.

So to make things clear, every time that you flip the coin, that’s considered to be a trial. There are 100 trials in a game (here $\colorbox[rgb]{1,1,1}{$N$}\colorbox[rgb]{1,1,1}{$=$}\colorbox[rgb]{1,1,1}{$100$}$). Suppose you played 50 games, … man you’re an addict. You might win $9 the first game, loose $12 the 2nd, loose $3 the 3rd, win $8 the fourth, etc. What do you make when you average over 50 games? What do you make when you average over an infinite number of games? That’s the left hand side of eqn. 1.15. Well how do we calculate the right hand side? Let’s take ${\colorbox[rgb]{1,1,1}{$x$}}_{\colorbox[rgb]{1,1,1}{$5$}}$. The first game it might be $1, the next game ${\colorbox[rgb]{1,1,1}{$x$}}_{\colorbox[rgb]{1,1,1}{$5$}}$ might be -$ 1, etc. So we want to average ${\colorbox[rgb]{1,1,1}{$x$}}_{\colorbox[rgb]{1,1,1}{$5$}}$ over an infinite number of games. Since you’re equally likely to win or loose, on average, this is 0. The same is true for all the ${\colorbox[rgb]{1,1,1}{$x$}}_{\colorbox[rgb]{1,1,1}{$i$}}$’s, so the right hand side is 0.

This says that on average, you expect to make 0 playing this game. The average was defined as one where you repeatedly play whole games over and over again, an infinite number of times.