Ergidicty is a concept in physics that equates the time average with the ensemble average of systems. In simple language, it means that the average property of a system remains the same by following one for several instances in time and averaging it (time average) or by averaging several states of the same at once (ensemble average). The first type is dynamic (changes with time), and the second is stochastic (statistical).
The assumption of ergodicity is fundamental to equilibrium statistical mechanics and, therefore, allows replacing dynamical descriptions with simpler probabilistic summaries. For example, the Brownian motion of gas molecules in a container is ergodic, meaning that a given molecule spends the same time in one half of the container as in the other half. In the coin tossing example, the average of results from tossing a fair coin infinite times equals tossing an infinite number of similar coins once.
Conversely, a stochastic process is non-ergodic when its statistics change with time. And this is what we saw in the special betting game.
