1

3

binomial distribution 43

according to the

GEOMETRIC SERIES

formula, is the

fraction .

Any

DYADIC

fraction has a finite binary decimal

expansion. As the study of those fractions shows, the

binary code of a dyadic can be cleverly interpreted as

instructions for folding a strip of paper to produce a

crease mark at the location of that dyadic fraction. The

process of R

USSIAN MULTIPLICATION

also uses binary

numbers in an ingenious manner.

Binary numbers are used in computers because the

two digits 0 and 1 can be represented by two alterna-

tive states of a component (for example, “on” or “off,”

or the presence or absence of a magnetized region).

It is appropriate to mention that the powers of 2

solve the famous “five stone problem”:

A woman possesses five stones and a simple

two-arm balance. She claims that, with a com-

bination of her stones, she can match the

weight of any rock you hand her and thereby

determine its weight. She does this under the

proviso that your rock weighs an integral num-

ber of pounds and no more than 31 pounds.

What are the weights of her five stones?

As every number from 1 through 31 can be represented

as a sum of the numbers 1, 2, 4, 8, and 16, the woman

has stones of weights corresponding to these first five

powers of 2.

binary operation A rule that assigns to each pair of

elements of a set another element of that same set is

called a binary operation. For example, the addition

of two numbers is a binary operation on the set of

real numbers, as is the product of the two numbers

and the sum of the two numbers squared. “Union” is

a binary operation on sets, as is “intersection,” and

CROSS PRODUCT

is a binary operation on the set of

vectors in three-space. However, the operation of

DOT

PRODUCT

is not a binary operation on the set of vec-

tors; the results of this operation are numbers, not

other vectors.

If the set under consideration is denoted S, then a

binary operation on Scan be thought of as a

FUNCTION

ffrom the set of pairs of elements of the set, denoted S

×S, to the set S: f : S ×S→S.

See also

OPERATION

;

UNARY OPERATION

.

binomial Any algebraic expression consisting of two

terms, such as 2x+ yor a+ 1, is called a binomial.

See also

MONOMIAL

;

POLYNOMIAL

;

TRINOMIAL

.

binomial coefficient See

BINOMIAL THEOREM

.

binomial distribution The distribution that arises

when considering the question

What is the probability of obtaining precisely k

successes in nruns of an experiment?

is called the binomial distribution. Here we assume the

experiment has only two possible outcomes—“success

or failure,” or “heads or tails,” for example—and that

the probability of either occurring does not change as

the experiment is repeated. The binomial distribution

itself is a table of values providing the answers to this

question for various values of k, from k= 0 (no suc-

cesses) to k= n(all successes).

To illustrate: the chance of tossing a “head” on a

fair coin is 50 percent. Suppose we choose to toss the

coin 10 times. Observe that the probability of attaining

any specific sequence of outcomes (three heads, fol-

lowed by two tails, then one head and four tails, for

example) is . In

particular, the probability of seeing no heads (all

tails) is also 1/1024, as is the chance of seeing 10

heads in a row.

There are 10 places for a single head to appear

among 10 tosses, thus the chances of seeing precisely

one head out of 10 tosses is , about

1 percent. According to the theory of

COMBINATION

s,

there are ways for two heads to

appear among 10 places, and so the probability of

seeing precisely two heads among the 10 tosses is

, about 4.4 percent.

Continuing this way, we obtain the binomial distri-

bution for tossing a fair coin 10 times: