Clearly, what is required here is a statistical treatment of the problem.
Instead of focusing on a single system, let us proceed in the usual
manner and consider a *statistical
ensemble* consisting of a large number of identical systems.
In general, these systems
are distributed over many different states at any given time.
In order to evaluate the probability
that the system possesses a particular property, we merely need to find the number of
systems in the ensemble which exhibit this property, and then divide by the total
number of systems, in the limit as the latter number tends to infinity.

We can usually place some general constraints on the system.
Typically, we know the total energy , the total volume , and the
total number of particles . To be more honest, we can only
really say that the total energy lies between and , *etc*., where
is an experimental error. Thus, we only need concern ourselves with
those systems in the ensemble exhibiting states which are consistent with the
known constraints. We call these the *states accessible to the system*.
In general, there are a great many such states.

We now need to calculate the probability of the system being found in
each of its accessible states. Well, perhaps ``calculate'' is the wrong word.
The only way we could calculate these probabilities would be to evolve all of the
systems in the ensemble and observe how long on average they spend in each
accessible state. But, as we have already mentioned, such a calculation is completely
out of the question.
So what do we do instead? Well, we effectively *guess* the
probabilities.

Let us consider an isolated system in *equilibrium*. In this situation,
we would expect
the probability of the system being found in one of its accessible
states to be independent of time. This implies that the statistical ensemble
does not
evolve with time. Individual systems in the ensemble will constantly
change state, but the
average number of systems in any given state should remain constant. Thus, all
macroscopic parameters describing the system, such as the energy
and the volume, should also remain constant. There is nothing in the laws of
mechanics which would lead us to suppose that the system will be found more
often in one of its accessible states than in another. We assume,
therefore, that *the system is equally likely to be found in any of
its accessible states*. This is called the assumption of *equal
a priori probabilities*, and lies at the very heart of statistical mechanics.

In fact, we use assumptions like this all of the time without really thinking
about them. Suppose that we were asked to pick a card at random from a well-shuffled pack. I think that most people would accept that we have
an equal probability of picking any card in the pack. There is nothing which
would favour one particular card over all of the others. So, since there are fifty-two cards in
a normal pack, we would expect the probability of picking the Ace of Spades, say, to
be . We could now place some constraints on the system.
For instance, we could only count red cards, in which case the probability
of picking the Ace of Hearts, say, would be , by the same
reasoning. In both cases, we have used the principle of equal *a priori*
probabilities. People really believe that this principle applies to games
of chance such as cards, dice, and roulette. In fact, if
the principle were found not to apply to a particular game most people would
assume that the game was ``crooked.'' But, imagine trying to prove
that the
principle actually does apply to a game of cards. This would be very difficult!
We would have to show that the way most people shuffle cards is effective
at randomizing their order. A convincing study would have to be
part mathematics and part psychology!

In statistical mechanics, we treat a many particle system a bit like an
extremely
large game of cards. Each accessible state corresponds to one of the
cards in the pack. The interactions between particles cause the system to
continually change state. This is equivalent to constantly shuffling the
pack. Finally, an observation of the state of the system is like picking
a card at random from the pack. The principle of equal *a priori*
probabilities then boils down to saying that we have an equal chance of
choosing any particular card.

It is, unfortunately, *impossible* to prove with mathematical rigor that the
principle of equal *a priori* probabilities applies to many-particle systems. Over the years, many people have attempted
this proof, and all have failed miserably.
Not surprisingly, therefore, statistical mechanics was greeted with a
great deal
of scepticism when it was first
proposed just over one hundred years ago. One of the its main proponents,
Ludvig Boltzmann, got so fed up with all of the criticism that he eventually
threw himself off a bridge!
Nowadays, statistical mechanics is completely accepted into the cannon
of physics. The reason for this
is quite simple: *it works!*

It *is* actually
possible to formulate a reasonably convincing scientific case for
the principle of equal *a priori* probabilities. To achieve
this we have to make use of the
so-called * theorem*.