and Statistics in the 18th History of Probability Centurypeople.math.umass.edu/~tevelev/475_2018/HPS.pdf · Jakob Bernoulli (1655-1705) ... Showed chance of success in an experiment

History of Probability and Statistics in the 18th

CenturyDeirdre Johnson, Jessica Gattoni, Alex Gangi

Jakob Bernoulli (1655-1705)

“The only thing needed for correctly forming conjectures on any matter is to determine the numbers of these cases accurately and then determine how much more easily some can happen than others. But here we come to a halt, for this can hardly ever be done. Indeed, it can hardly be done anywhere except in games of chance… But what mortal, I ask, may determine, for example the number of cases, which may invade at any age the innumerable parts of the human body and which imply our death? And who can determine how much more easily one disease may kill than another?... Likewise who will count the innumerable cases of the changes to which the air is subject every day and on this basis conjecture its future constitution after a month, not to say after a year?”

Ars ConjectandiPublished in 1713 by nephew Niklaus Bernoulli

Divided into four parts

1. Commentary added to Christian Huygen’s De Ratiociniss in aleae ludo (On the Calculations in Games of Chance)

2. Further developed laws of permutations and combinationsa. Generalized some of Pascal’s ideas about division of stakes in an “interrupted game”

i. Showed chance of success in an experiment in which chance of success and chance of failure are not equal

b. Introduced Bernoulli Numbers3. Applications to games of chance 4. The Use and Application of the Preceding Doctrine in Civil, Moral, and Economic

Mattersa. Moral certainty and morally impossibleb. Law of Large Numbers

Law of Large Numbers: Activity

In a deck of 35 cards, there are x face cards (not including aces).

With replacement, pull out a card one at a time.

What can we estimate as the probability of pulling a face card after 5 trials?

10?

15?

20?

Law of Large NumbersGoal: To show that “as the number of observations increases, so the probability increases of obtaining the true ratio between the number of cases in which some event can happen and not happen, such that this probability may eventually exceed any given degree of certainty.”

Bernoulli: Given any small fraction and any large positive number c, a number N=N(c) may be found so that the probability that X/N differs from p by no more than ε is greater than c times the probability that X/N differs from p by more than ε.

Bernoulli focused on determining the value of N(c) to get the true probability

Modern: Given any and any positive number c, there exists an N such that

Finding Probability with Moral Certainty

To find N such that true probability can be found with “moral certainty”, Bernoulli set c = 1000. He then showed that if t = r+s, then N(c) could be taken as a larger integer than the greater of:

and

where m,n are integers such that

and

Find N(c) using our information from the activity

Abraham De Moivre (1667-1754)

● Born in Vitry, France● Born into a Protestant family● Age 11-14: educated in classics at Protestant secondary school

in Sedan● Later, studied in Saumur (Huygen) and then in Paris (physics

and standard mathematics)● After revocation of edict of Nantes (1685), life was difficult in

France○ Imprisoned from about 1686-1688

● April 1688: left France to go to England● 1697: Elected into the Royal Society

○ Never received a university position● Made a living by tutoring and solving problems from games of

chance and annuities

The Doctrine of Chances● Published in 1718 by De Moivre

○ New editions in 1738 and 1756● Probability: “The Probability of an Event is greater, or less, according to the number of

Chances by which it may happen, compared to the whole number of Chance by which it may either happen or fail”

Problem III

“To find in how many trials an event will probably, happen, or how many trials will be necessary to make it indifferent to lay on its happening or failing, supposing that a is the number of chances for its happening in any one trial and b the number of chances for its failing.”

Solution: Application:1. How many throws of two are

necessary to give even odds of throwing two sixes?

2. How many times must you pull 4 cards out of a standard deck to give even odds of pulling three aces?

The Doctrine of Chances: 2nd Edition● In 2nd edition, published 1738, De Moivre included a brief paper he wrote titled

“Approximatio ad Summam Terminorum Binomii in Seriem expansi”○ Gave first statement for formula of the normal curve○ Gave “first method of finding the probability of an occurrence of an error of a given size when

that error is expressed in terms of of the variability of the distribution as a unit”○ Gave first mention of “probable error”

● Approximated the sums of terms of the binomial ○ Goal: to estimate the probability by using experiments

Application of Probability: Annuities

● Considered a “bet” by the annuitant and a loan at interest by the seller○ Paid regular payments until his death○ Essentially betting that he would live long enough to collect his payments

and then some ● This is used the same today with life insurance companies

○ Today life insurance provides provides financial protection for a specific period of time, after that period is up, if you are still alive, the insurance company will opt to raise the price of the insurance.

Thomas Bayes (1702-1761)

● Born into a Nonconformist, wealthy family● Possible he was privately tutored by De Moivre● Studied logic and theology at the University of Edinburgh

○ No written record of studying mathematics● Ordained as a Nonconformist minister● Published little of his own work

○ Most of his work was published after his death● His work was often critiquing and expanding on his peers work

○ One such work was: Essay towards solving a problem in the doctrine of chances

● 1742: Elected into Royal Society despite having no publications● There is little primary sources of Bayes life, so there is a lot of

speculation of some parts of his life.

Essay towards solving a problem in the doctrine of chances

“I now send you an essay which I have found among the papers of our deceased friend Mr. Bayes, and which, in my opinion, has great merit... In an introduction which he has writ to this Essay, he says, that his design at first in thinking on the subject of it was, to find out a method by which we might judge concerning the probability that an event has to happen, in given circumstances, upon supposition that we know nothing concerning it but that, under the same circumstances, it has happened a certain number of times, and failed a certain other number of times.”

Richard Price, Member of Royal Society

Bayes TheoremBayes work in Essay towards solving a problem in the doctrine of chances is better known to us as Bayes Theorem.

Bayes writes it as:

“ If there be two subsequent events, the probability of the 2nd b/N and the probability of both together P/N, and it being first discovered that the 2nd event has happened, from hence I guess that

the 1st event has also happened, the probability I am in the right is P/b.”

This theorem deals with conditional probability. Conditional probability is the probability of an event happening, given that it has some relationship to one or more other events.

Bayes TheoremWith this formula, we are trying to figure out the probability that A occurs given that B has occurred.

Bayes Theorem has a few forms that are useful for different problems, but they are all equivalent.

Problem with Bayes TheoremIn a factory there are two machines manufacturing bolts. The first machine manufactures 55% of the bolts and the second machine manufactures the remaining 45%. From the first machine 8% of the bolts are defective and from the second machine 4% of the bolts are defective. A bolt is selected at random, what is the probability the bolt came from the first machine, given that it is defective?

Let A be the event that a bolt came from Machine 1 and let B be the event that a bolt is defective.

Let’s define the probabilities:

P(A) = .55 P(A’) = .45 P(B|A) = .08 P(B|A’) = .04

Now we use the second form of Bayes Theorem from the previous slide.

(0.08 × 0.55)/((0.08 × 0.55) + (0.04 × 0.45)) = .7097

There is a 70.97% chance that given the selected bolt is defective, it is from the first machine.

Class Problems with Bayes Theorem1. A study shows that 2.5% of women over 50 have breast cancer. It also says

86% of women who have breast cancer test positive on mammograms. However, 8.9% of women will have false positives. What is the probability that a woman has cancer if she has a positive mammogram result?

2. A genetic defect is present in 6% of people. A study shows 92% of tests for the gene detect the defect (true positives). It also shows that 7.3% of the tests are false positives. If a person gets a positive test result, what are the odds they actually have the genetic defect?

3. In a small shop there are two workers packaging the product. The first worker packages 58% of the product and the second worker packages the other 42%. The first worker mispackages 7% of the product and from the second worked mispackaged 3% of the product. A single product is randomly selected, what is the probability the product came from the second worker, given that it is mispackaged?

Answers

1. .1987 or 19.78% chance that a woman over 50 has cancer given she tested positive

2. .4458 or 44.58% chance of the person has the genetic defect given they tested positive

3. .2368 or 23.68% chance the selected product came from the 2nd worker given it was mispackaged

Modern Applications of Bayes Theorem

Bayes Theorem is still used in many situations today, as you saw with the problems.

It can be used for:

● Spam filtering● Weather Forecasting● Medicinal science

Works Cited● Katz, Victor J. A History of Mathematics: An Introduction. Addison-Wesley, 1998.

● https://books.google.com/books/about/Ars_conjectandi.html?id=kD4PAAAAQAAJ - picture slide 3

● De Moivre, Abraham. The Doctrine of Chances: or, A Method of Calculating the Probabilities of Events

in Play. 3rd ed., 1756. - picture slide 10

● http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bayes.html

● Smith, David Eugene. A Source Book in Mathematics. New York : McGraw-Hill Book Co., 1929.

● https://www.maa.org/press/periodicals/convergence/sums-of-powers-of-positive-integers-jakob-bernoull

i-1654-1705-switzerland - sum of integer powers

● https://mathbitsnotebook.com/Geometry/Probability/PBBinomialProbNormalCurve.html picture on

slide 11