J. Bernouli (1654-1705) discovered the law of large numbers.
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Step-by-step explanation:
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Answer:
The Bernoulli family is the most
prolific family of western mathematics.
In a few generations in the 17th and 18th
centuries more than ten members
produced significant work in
mathematics and physics. They are all descended from a Swiss family of successful spice traders. It is
easy to get them mixed up (many have the same first name.) Here are three rows from the family tree. The
six men whose boxes are shaded were mathematicians.
For contributions to probability and statistics, Jacob Bernoulli (1654-1705)
deserves exceptional attention. Jacob gave the first proof of what is now called the
(weak) Law of Large Numbers. He was trying to broaden the theory and
application of probability from dealing solely with games of chance to “civil, moral,
and economic” problems.
Jacob and his younger brother, Johann, studied mathematics against the wishes of
their father. It was a time of exciting new mathematics that encouraged both
cooperation and competition. Many scientists (including Isaac Newton and Gottfried Leibniz) were
publishing papers and books and exchanging letters. In a classic case of sibling rivalry, Jacob and Johann,
themselves, were rivals in mathematics their whole lives. Jacob, the older one, seems to have been
threatened by his brilliant little brother, and was fairly nasty to him in their later years.
Here is a piece of a letter (1703) from Jacob Bernoulli to Leibniz that reveals the tension between the
brothers. Jacob seems worried that Johann, who had told Leibniz about Jacob’s law of large numbers,
may not have given him full credit.
I would very much like to know, dear sir, from whom you have it that a theory of estimating probabilities has been
cultivated by me. It is true that several years ago I took great pleasure in this sort of speculations, so that I could hardly
have thought any more about them. I had the desire to write a treatise on this matter. But I often put it aside for whole
years because my natural laziness compounded by my illnesses made me most reluctant to get to writing. I often wished
for a secretary who could easily understand my ideas and put them down on paper. Nevertheless, I have completed most
of the book, but there is lacking the most important part, in which I teach how the principles of the art of conjecturing
are applied to civil. moral. and economic matters. [This I would do after] having solved finally a very singular problem, of
very great difficulty and utility. I sent this solution already twelve years ago to my brother, even if he, having been asked
about the same subject by the Marquis de l'Hopital, may have hid the truth, playing down my work in his own interest.”
(April, 1703)
Jacob died in 1705, leaving many
unpublished papers. Eight years later,
in 1713, his family published Jacob’s
notes for the book he mentioned in
the letter to Leibniz, Ars Conjectandi
(The art of conjecturing). The book
fundamentally changed the way
mathematicians approached
probability. Its crowning achievement
was the detailed proof of Jacob’s law
of large numbers.
Jacob Bernoulli
Vermont Mathematics Initiative, Bob Rosenfeld
2
What is Bernoulli’s Law of Large Numbers?
Jacob’s theorem is what we informally call the Law of Averages. It says that if you take more and more
observations of some random outcome (such as the sum of two dice), then the mean of all your
observations will eventually approach the theoretical Expected Value and from that point stay very close
to it. The label, “Law of Large Numbers,” first appeared in 1837, more than 100 years after Bernoulli
proved it, in a book by the French mathematician, Simeon Denis Poisson, who proved a more general
version of the theorem.
Things of every kind of nature are subject to a universal law which one may well call the Law of Large
Numbers. It consists in that if one observes large numbers of events of the same nature depending
on causes which are constant and causes which vary irregularly, . . . , one finds that the proportions
of occurrence are almost constant . . .[ [In the original French, Poisson wrote Loi des grands nombres.]
Here’s typical graph, based on a computer simulation of
300 rolls of two dice. Recall, the expected value for the
sum of two dice is 7. The graph should be approaching
height 7. This image makes it clear that “eventually”
may mean a really long time.
In this simulation the mean was 6.82 after 300 rolls. It
was off from the expected value by 0.18. True, it was
no longer wildly away from 7 like it was at the
beginning, but suppose you need to be “sure” (Bernoulli
said “morally certain”) that your experimental mean
was off from the true expected value by no more than
0.01. The law of large numbers gives you a minimum
value for the number of rolls, n, to accomplish this.