Probability

Some lessons from history

Girolamo Cardano 1501-1576• Trained as a medical doctor but

found it hard to earn enough• Addicted to gambling – lost all his

money and ended up in the poorhouse

• Got a job as a mathematics lecturer

• Best known for his work on solving cubic and quartic equations.

The first book about probability• Cardan wrote the first book about probability • Published after his death • “How many throws of a fair die do we need in

order to have an even chance of at least one 6?”• Cardan thought it should be 3 throws as that is

half the total number of faces on the die• Was he right?

Galileo Galilei 1564 - 1642• Taught mathematics from

the 1580s• Became Mathematician and

Philosopher to the Grand Duke of Tuscany in 1610

• Under house arrest from 1633 for the heretical opinion that the sun was the centre of the universe

The game of passedix• It is one of the most ancient games of chance.• It is played with three dice.• The three scores are added up.• If a player throws less than 10, the banker wins

the money of all players.• If a player throws more than 10, the banker

doubles the money of all players.• Rules vary for a total of exactly 10.

The Duke of Tuscany’s question• 9 and 10 can each be obtained in the same

number of ways so why does experience show that getting a total of 10 is more likely?

Total of 9 1+2+6 1+3+5 1+4+4 2+2+5 2+3+4 3+3+3

Total of 10 1+3+6 1+4+5 2+2+6 2+3+5 2+4+4 3+3+4

An extension question• In the game of Passedix, which is a fairer way to

play?– Total of less than 10 v 10 or more– Total of 10 or less v more than 10

The Chevalier de Mere• In the 17th century, a French nobleman, the

Chevalier de Mere, played two different games of chance.

• Rolling at least one 6 in four throws of a single die

• Rolling at least one double 6 in 24 throws of a pair of dice.

The Chevalier’s reasoning 1654On one throw of a die,

Average number of 6s in four throws =

Throwing two dice,

Average number of double 6s in 24 rolls =

1P(six) =6

1 246 3

× =

1P(double six) =36

1 22436 3

× =

Why did he lose more often on the second game?

The Chevalier wrote to his friend Pascal who solved the problem with his friend Fermat.

Letter from Pepys to Newton 1693

• Pepys was planning to make a wager so he wrote to Newton for advice

• Which is more likely?– At least one 6 when rolling six dice– At least two 6s when rolling twelve dice– At least three 6s when rolling eighteen dice

Jean Le Rond d’Alembert 1717 - 1783

• d’Alembert pioneered the study of partial differential equations

• He also contributed to a 28 volume encyclopaedia, writing most of the mathematical articles

d’Alembert’s encyclopedia 1754

Bertrand’s box paradox 1889• There are three boxes with two drawers each.

– One box has a gold coin in each drawer.– One box has a gold coin in one drawer and a

silver coin in the other.– One box has a silver coin in each drawer

• A box is chosen at random and a drawer is opened at random. The coin in the drawer is gold. What is the probability that the coin in the other drawer is gold?

A question about 3 cardsI have 3 cards:

yellow on both sides

black on both sides

yellow on one side, black on the other

I shuffle and turn the cards thoroughly and pick one without looking then show you one side of it. It is yellow. What is the probability that the card is yellow on both sides?

The Work of Tversky and Kahneman

• Amos Tversky and Daniel Kahneman have published a number of research papers on people’s understanding of probability.

• Amos Tversky died 1996.• Kahneman published a best-selling book in 2011

summarising their research: Thinking fast and slow

• Kahneman has a Nobel prize for economics

How many would you expect?In 4 pages of a novel (about 2000 words) how many words would you expect to find of the form ----ing? (i.e. 7 letter words ending –ing)

How many would you expect?In 4 pages of a novel (about 2000 words) how many words would you expect to find of the form -----n-? (i.e. 7 letter words with n in next to last place)

The results• A group of university students estimated an

average of 6.4 words of form ----ing and 2.9 words of form -----n-

Which is more likely?

• ----ing • Or• -----n-

Gerd Gigerenzer’s workA 50-year-old woman, no symptoms, participates in routine mammography screening. She tests positive, is alarmed, and wants to know from you whether she has breast cancer for certain or what the chances are. Apart from the screening results, you know nothing else about this woman. How many women who test positive actually have breast cancer? What is the best answer?

nine in 10 eight in 10one in 10 one in 100

US statistics from 1990s• The probability that a woman has breast cancer

is 1% ("prevalence")• If a woman has breast cancer, the probability

that she tests positive is 90% ("sensitivity")• If a woman does not have breast cancer, the

probability that she nevertheless tests positive is 9% ("false alarm rate")

Gigerenzer’s research• He asked over 1000 gynaecologists• 21% gave the correct answer • the correct answer is “one in ten”

The use of natural frequencies

1000 women10 have cancer 990 don’t have

cancer9 test

positive1 tests

negative89 test positive

901 test negative

NHS Mammography leaflet

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