where Dis the diameter of the circumcircle. (Here we
make use of the fact that any angle subtended from a
diameter is 90°.) Thus the three quantities expressed in
the law of sines each equal the diameter of the circum-
circle of the triangle, thereby offering an alternative
proof of the law.
See also
CIRCLE THEOREMS
.
law of large numbers If one tosses a fair coin 10
times, one would expect, on average, five of those
tosses to be “heads.” Of course, in any single run of 10
tosses, any number of heads is possible, even a string of
10 heads in a row, but this is extremely unlikely. The
number of heads actually observed in an experiment is
likely to be four, five, or six, close to 50 percent. (The
probability of any particular count of heads appearing
is described by the
BINOMIAL DISTRIBUTION
.)
In a run of 100 tosses we would expect the effects
of excessive runs of heads, or tails, to “average out”
and the proportion of heads obtained to be even closer,
on average, to the “true value” of 50 percent—and
closer still if we run an experiment of 1,000 or 10,000
tosses, or more. This is the law of large numbers in
action. Precisely, this law states the following:
The more times a random phenomenon is per-
formed, the closer the proportion of trials in
which a particular outcome occurs approxi-
mates the true probability of that outcome
occurring.
If, for example, a 1 never occurred when rolling a die
10 times, we can be assured, however, that the propor-
tion of 1s appearing in another 100, 1,000, 10,000, …
tosses will approach the value one-sixth.
Many gamblers incorrectly interpret the law of
large numbers as a method for predicting outcomes of
random events. (See
LAW OF AVERAGES
.)
The law of large numbers is a mathematical conse-
quence of C
HEBYSHEV
’
S THEOREM
. It can be interpreted
as saying that if a random phenomenon produces
numerical outcomes with mean value µ, then the mean
of Nobserved values of the phenomenon approaches
the value µas Nincreases. Chebyshev’s theorem is
related, for it gives measures of how values are dis-
tributed about the mean.
See also M
ONTE
C
ARLO METHOD
;
STATISTICS
:
DESCRIPTIVE
.
law of sines (sine rule) See
LAW OF COSINES
/
LAW
OF SINES
.
law of the lever A
RCHIMEDES OF
S
YRACUSE
(ca.
287–212
B
.
C
.
E
.) recognized that two weights w1and w2
placed at distances x1and x2, respectively, from the ful-
crum (pivot point) of a simple lever will balance when
x1w1= x2w2. This principle is called the law of the
lever. For example, an adult weighting twice as much
as a child will balance on a seesaw if she sits half the
distance from the pivot point as the child.
See also
CENTER OF GRAVITY
.
laws of thought The Greek philosopher A
RISTOTLE
(384–322
B
.
C
.
E
.) identified three laws of logic, all tau-
tologies (meaning that each has a
TRUTH TABLE
with
constant value T) that have since been deemed funda-
mental descriptions of the way we think. His three laws
of thought are:
1. Law of Noncontradiction: It is not the case that
something can be both true and not true.
Symbolically:
¬
[p(
¬
p)]
2. Law of Excluded Middle: Each must either be true
or not true.
Symbolically: p∨(
¬
p)
3. Law of Identity: If something is true, then it is
true.
Symbolically: p→p
Mathematicians often rely on the law of excluded
middle to establish the validity of mathematical results:
an
INDIRECT PROOF
or a
PROOF BY CONTRADICTION
proves that a statement pis true by showing that it
cannot be false. However, not all philosophers (and
mathematicians) agree with this approach and question
the validity of this second law. For example, as the
20th-century Austrian mathematician K
URT
G
ÖDEL
showed, there are some statements in mathematics that
can neither be proved nor disproved, and are conse-
quently neither true nor false. The constructivist move-
ment accepts results established by
DIRECT PROOF
only.
To move beyond the law of the excluded middle,
logicians have attempted to generalize
FORMAL LOGIC
to include three possible values of truthhood: true,
∨
304 law of large numbers