CHAPTER V
THE PROBABILITIES IN THE PROBLEM

To help us towards a solution of this paradox, we must
first find out what the probabilities actually lead us to
conclude. We have already seen that, in a given case, the
chances are even as to whether energy will run down or
build up. There are also small chances of a neutral
condition,in which energy remains, on the whole, at
the same difference of concentration as before. But the
probability of this neutrality is negligible, and we may say
that the probabilities are, that in 50% of the cases the
second law of thermodynamics will be obeyed, and in 50%
of the cases it will be reversed. If such is the case, the
universe as a whole will be neutral; that is, taking all the
occurrences over all of time and space, there will be no
tendency in one direction or the other.
In this reasoning we can be assured as to the
probabilities in any given occasion, for we must assume all
combinations of initial positions and velocities to be equally
likely. Inasmuch as any event occupies a certain amount of
time, let us figure on the probabilities of the positions and
initial velocities at. the middle of that interval. For any range
of positions and velocities resulting in a combination
obeying the second law of thermodynamics, we have an
equal and therefore an equally probable range of positions
and velocities reversing that law; namely, the identical
positions with the reverse velocities. Where the positions
and velocities happen to border between the two kinds of
combinations. we will have a sort of neutral result, which
is so improbable as to have a zero probability (though that
does not make it impossible). Aside from that, the second
law of thermodynamics is, on any occasion, equally
probable with its reverse, and the probability of each may
be taken as 50%. The probability of the second law of
thermodynamics being followed on two certain occasions
is, as a result, only 25%; and so on, while its probability for
all occasions is almost a nullity.
The probability is. however. as a result of this 50%
probability, that approximately half the events of the
universe, taking all of space and time, will be in accordance
with the second law of thermodynamics, while about half will
tend to reverse it. The former tendency we will, for short, call
the positive, while the latter tendency we will call the
negative tendency. Between these two there is a bordering,
or neutral, tendency, which, as a whole, neither builds
energy up nor levels it down.
The universe as a whole, Including all of time and
space, will tend towards this neutral tendency, but this
neutral tendency will simply be a compound of positive and
negative tendencies at different parts of space and time
tending to cancel one another. Taking definite portions of
space and time, the chances are that there will be some sort
of preponderance of tendency in one direction or the other,
the preponderance being greater the smaller the section of
space and time that we take Into consideration. We may,
therefore, assume that, in the part of space and time under
our observation (which, we know is very limited) the
preponderance is towards the positive tendency. We may
suppose that there are other parts of space, and other
periods of time, when the preponderance will be in the
reverse direction.
But even where the preponderance Is toward the postive
tendency; It still remains' merely a preponderance,

and instances of the negative tendency would be almost
certain to occur. it is true that the probabilities are that, in
such a part of space and time, instances of the negative
tendency will occur to a very limited amount; but, all the
same, they will occur.
The probabilities of the situation, then, are as follows: 'the
whole universe, including all of space and time, will tend to
have as much of the positive as of the negative tendency.
with a certain amount of the neutral tendency. At a particular
moment of time the probabilities are that there will also he
about as much of one tendency as of the other, but that in
some sections of space there will be a preponderance
towards the positive tendency, wbile in other sections of
space the preponderance will be the reverse; about half of
space falling under one heading. and about half of space
falling under the other. In each of those portions of space
there will be instances of events opposed to the prevailing
tendency, presumably in certain material objects. The same
applies to a more limited extent if we take one section of
space with respect to the different moments of time.