The mirror. (Stillwater, Minn.) 1894-1925, 1901-07-18
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case of coin tossed in the air: When
it falls, either head or tail must be
uppermost; and just as the geometer
in beginning demonstration says:
"Let this be circle, or suppose this to
be circle" so we suppose an ideal
case, namely, that the conditions which
determine whether the coin falls with
its head or tail uppermost are such as
to render either event equally probable.
Then whatever symbol is to represent
the chance of head will also represent
that of a tail. What shall this symbol
be? 1t must evidently express the
! magnitude of the chance; it is a
question of more or less: l. O., of
measurement, the chances being in the
supposed case equal and therefore the
| measure of the same.
Now measurement implies a unit.
To measure probability we must have
unit or standard, just as to measure
distance we must have a unit of
length, and the one essential quality of
a unit is its invariableness; a yard
stick which is continually changing it's
length is of no value. Our unit then
must be constant. Such unit is a
certainty. What this is, we all under
| stand, and certainty is our unit of
probability; so that if a unit expresses
certainty, a probability will be
fractional part of unity. Thus, in the
case of the coin, the chance of a head
is 4, of tail is Il, and 4 and 4 is 1;
| that is, either head or tail must turn
up, and we have the sum of probabilities
equal to unity, or certainty, as it should
| be.
Take another ideal case: Seven black
balls and three white balls are in an
urn, and each is equally liable to be
drawn. Then the probability of draw
ing black ball is 7-1O, and drawing a
white one is $40, and 7-1O and $40 is 1,
or we are certain to draw either black
ball or white one. We see, then, that
the measure of probability of the
happening of an event "is expressed
by the fraction that the number of
favorable cases is to all that can
happen"
Now take our ideal case into practice.
| In practice the chances of a head or
tail in single throw are equal. There
are multitude of conditions deter
mining how the coin is to fall, about
which we have no knowledge, and over
which we have no control. They may
be such that the head falls uppermost
several times in succession, so that our
mathematical statement that the
probability of tail is 4 seems good
for nothing. Here enters experience.
AS matter of fact, we find in spite of
all these varying conditions under
which the coin falls, and our ignorance
of them, in the long run it falls on one
side about as often as on the other, or
that in many trials the conditions tend
to equality. In short, experience
teaches us that while our statement of
an equal number of heads and tails in
six throws is of little value, in 100 it is
of more value; and while that for no
definite number of trials, however
great, can we predict what the result
will be, we can make general statements
which actual trial will in the long run
justify, and which, like general prin-
ciples in other matters, are in the long
run safe rules to follow. So it is with
the balls. In ten trials we should
rarely get seven black and three white
ones. Indeed the probability would be
very great against it. But in 1000 trials
the proportion of the number of black
to white balls would be nearly seven to
three. Why? Because experience so
teaches us. MAHNUD.