Bernoulli family 39

Defense of the Mathematicians Against the Objections

of the Author of The Analyst,” defending the logical

foundations of S

IR

I

SAAC

N

EWTON

’s newly invented

CALCULUS

. Despite the apparent lack of published math-

ematical work, Bayes was nonetheless elected a fellow

of the prestigious academic R

OYAL

S

OCIETY

in 1742.

Bayes retired from the ministry in 1752 but

remained in Tunbridge Wells until his death on April

17, 1761. His friend, Richard Price, discovered the

now-famous paper on probability theory among his

belongings and submitted it for publication. A second

paper, “A Letter on Asymptotic Series from Bayes to

John Canton,” one on asymptotic series, was also pub-

lished after Bayes’s death. The theoretical approach of

inferential statistics Bayes proposed remains an active

area of research today.

See also B

AYES

’

S THEOREM

;

STATISTICS

:

INFERENTIAL

.

Bayes’s theorem In his 1763 paper, published

posthumously, R

EV

. T

HOMAS

B

AYES

established a fun-

damental result, now called Bayes’s theorem, that

expresses the

CONDITIONAL PROBABILITY

P(A

|

B) of an

event Aoccurring given that event Bhas already

occurred in terms of the reverse conditional probability

P(B

|

A). Precisely:

This formula is easily proved by noting that

and .

More generally, suppose B1, B2,…, Bnis a mutually

exclusive and exhaustive set of events, that is, a set of

nonoverlapping events covering the whole

SAMPLE

SPACE

. Suppose also that we have been told that

another event Ahas occurred. Then the probability

that event Bialso occurred is given by:

To illustrate: suppose that bag 1 contains five red balls

and two white balls, and bag 2 contains seven red balls

and four white balls. If a bag is selected at random and

a ball chosen from it is found to be red, what is the

probability that it came from bag 1?

Here let Abe the event “a red ball is chosen” and B1

and B2the events “a ball is selected from bag 1 / bag 2,”

respectively. Then P(B1) = 1/2 = P(B2), P(A

|

B1) = 5/7,

and P(A

|

B2) = 7/11. Thus the probability we seek,

P(B1

|

A), is given by:

bearing The

ANGLE

between the course of a ship and

the direction of north is called the ship’s bearing. The

angle is measured in degrees in a clockwise direction

from north and is usually expressed as a three-digit

number. For example, a ship heading directly east has a

bearing of 090 degrees, and one heading southwest has

a bearing of 225 degrees.

The word “bearing” is also used for the measure of

angle from north at which an object is sighted. For

example, a crewman on board a ship sighting a light-

house directly west will say that the lighthouse has

bearing 270 degrees.

Bernoulli family No family in the history of mathe-

matics has produced as many noted mathematicians as

the Bernoulli family from Basel, Switzerland. The family

record begins with two brothers, Jacob Bernoulli and

Johann (Jean) Bernoulli, respectively, the fifth and 10th

children of Nicolaus Bernoulli (1623–1708).

Jacob (December 27, 1654–August 16, 1705) is

noted for his work on

CALCULUS

and

PROBABILITY

the-

ory, being one of the first mathematicians to properly

understand the utility and power of the newly pub-

lished work of the great W

ILHELM

G

OTTFRIED

L

EIB

-

NIZ

(1646–1716). Jacob applied the calculus to the

study of curves, in particular to the logarithmic spiral

and the

BRACHISTOCHRONE

, and was the first to use

POLAR COORDINATES

in 1691. He also wrote the first

text concentrating on probability theory Ars con-

jectandi (The art of conjecture), which was published