Mathematical Ideas that Shaped the World

1
Mathematical Ideas that Shaped the World

• Bayesian Statistics

2
Plan for this class

• Why is our intuition about probability so bad?
• What is the chance that two people in this room
  were born a few days apart?
• What is conditional probability?
• If someones DNA is found at a crime scene, what
  is the chance they are guilty?
• How can we spot bad statistics in the media?

3
An unfortunate truth

• Humans have an extraordinarily bad intuition
  about probability.

4
Winning the lottery

• What do you think your chances of winning the
  lottery are?
• Say whether winning the lottery is more or less
  likely to happen than this collection of events

5
Is winning the lottery more or less likely?
LESS
MORE
1 in 4,096

• Chance of getting 12 heads in a row when flipping
  a fair coin.

6
Is winning the lottery more or less likely?
LESS
MORE
1 in 24,000

• Dying from a road accident in 1 year

7
Is winning the lottery more or less likely?
LESS
MORE
1 in 25 million

• Dying in the next flight you take

8
Is winning the lottery more or less likely?
LESS
MORE
1 in 1 million

• Being struck by lightning

9
Is winning the lottery more or less likely?
LESS
MORE
1 in 300 million

• Dying from a shark attack

10
Is winning the lottery more or less likely?
LESS
MORE
1 in 2 million

• Dying in the next hour from any causes whatsoever

11
Conclusion

• Winning the lottery has surprisingly bad odds 1
  in 13,983,816.
• Yet many people are convinced that this could one
  day be likely to happen to them.
• We mix up the probability of someone winning the
  lottery (which is quite likely) with the
  probability of us winning the lottery.

12
The birthday problem

• How many people need to be a room together so
  that there is a more than 50 chance of two
  people having the same birthday?

A) 300
B) 183
C) 91
D) 23
13
Number of people Probability that 2 people share a birthday
10 11.7
20 41.1
23 50.7
30 70.6
50 97
57 99
100 99.99997
200 99.9999999999999999999999999998
366 100
14
The birthday graph
15
In this room?

• What is the chance that two people in this room
  have birthdays less than 3 days apart (ignoring
  the year?)

Answer more than 50
16
Monty Hall

• Behind 1 door is a sheep. Behind the other 2
  doors are other, non-sheepy, animals.
• You choose a door. I open a different door
  showing a non-sheep.
• Given the choice now of sticking with your choice
  or switching, what should you do?

17
Suppose you choose Door 1
Door 1 Door 2 Door 3 Stick Switch
Sheep! Not a sheep Not a sheep Sheep! No sheep
Not a sheep Sheep! Not a sheep No sheep Sheep!
Not a sheep Not a sheep Sheep! No sheep Sheep!
If you stick with your choice, you only win 1
time out of 3.
18
Conditional probability

• Conditional probability is the chance of
  something happening given that another event has
  already happened.
• For example you throw two dice. What is the
  probability of the first die being a 6 given that
  the sum of the two dice is 8?
• What if the sum of the two dice was 6 or 7?

19
How to think about conditional probability

• Conditional probability is all about updating
  your odds in light of new evidence.
• There are a priori odds the initial probability
  of an event.
• E.g. the probability of rolling a 6 is a priori 1
  in 6.
• After new evidence, you have a posteriori odds.
• E.g. the probability of having a 6, given that
  the sum of two dice is 8, is 1 in 5.

20
Boy or girl?

• I know a friend who has 2 children.
• At least one of the children is a boy.
• What is the chance that the other child is also a
  boy?

Answer 1 in 3
21
Explanation

• A priori, there are 4 possible combinations of
  children
• Boy Boy
• Boy Girl
• Girl Boy
• Girl - Girl
• From our new evidence, we know that Girl-Girl is
  not possible, leaving only 3 options.
• Of these 3 options, only one of them is Boy-Boy.

22
A paradox?

• If you know that the oldest child is a boy, the
  probability of the other child being a boy is
  50.
• If you know that the youngest child is a boy, the
  probability of the other child being a boy is
  50.
• Surely the first boy must be either the youngest
  or the oldest?!

23
Homework

• I know a friend who has two children.
• At least one of the children is a boy who was
  born on a Tuesday.
• What is the chance that the other child is also a
  boy?

24
Confusion of the inverse

• People have a tendency to assume that a
  conditional probability and its inverse are
  similar. For example
• If sheep enjoy eating grass, then an animal who
  likes grass is likely to be a sheep.
• If most accidents happen within 20 miles of home,
  then you are safest when you are far from home.

25
Manipulating statistics

• A. Taillandier (1828) found that 67 of prisoners
  were illiterate.
• What stronger proof could there be that
  ignorance, like idleness, is the mother of all
  vices?
• But what proportion of illiterate people were
  criminals?

26
Bayesian statistics

• The first person we know who looked seriously
  into conditional probabilities was Thomas Bayes.
• He was the first person to write down a formula
  connecting the two inverse conditional
  probabilities.
• Bayesian statistics is all about updating the
  odds of an event after receiving new evidence.

27
Thomas Bayes (1702 1761)

• Son of a London Presbyterian minister.
• Studied logic and theology at the University of
  Edinburgh.
• In 1722 returned to London to assist his father
  before becoming a minister of his own church in
  Tunbridge Wells, Kent, in 1733.

28
Thomas Bayes (1702 1761)

• During his lifetime, Bayes only published two
  papers.
• One was on Divine Benevolence.
• The other was a defence of The Doctrine of
  Fluxions against the attack of George Berkeley.
• His most famous paper was published in 1764,
  called An Essay towards solving a problem in the
  Doctrine of Chances.

29
Bayes Theorem

• P(A) is the prior probability of A.
• P(B) is the prior probability of B.
• P(AB) is the probability of A happening, given
  that B has happened.
• P(BA) is the probability of B happening, given
  that A has happened.

30
Importance of Bayes Theorem

• Bayes Theorem is especially useful in medicine
  and in law.
• Most doctors get the following question wrong.
  Lets see what you think!

31
A test for breast cancer

• 1 of women aged 40 will get breast cancer.
• Out of the women who have breast cancer, 80 of
  them will have a positive test result.
• Out of the women who dont have breast cancer,
  10 of them will get a positive result.
• If a woman tests positive for breast cancer, what
  is the chance she has actually has it?

32
Doing the numbers

• Consider 10,000 women.
• 100 of them will have breast cancer.
• 80 of them test positive
• 20 of them test negative
• 9900 of them dont have breast cancer.
• 990 of them test positive
• 8910 of them test negative
• In total there are (80990) 1070 positive
  results, of which only 80 have cancer.
• Thats 7.4.

33
The prosecutors fallacy

• Suppose a prosecutor in a court case finds a
  piece of evidence e.g. a DNA sample.
• They argue that the probability of finding this
  evidence if the defendant were innocent is tiny.
• Therefore the defendant is very unlikely to be
  innocent.
• Where is the fallacy in this argument?

34
The prosecutors fallacy

• If the a priori chance of the defendants guilt
  is very low, then it will still be very low after
  presentation of this evidence.
• Just like with the cancer example, a false
  positive may be much more likely than a true
  positive in the absence of other evidence.

35
Exhibit 1 Sally Clark, 1999

• Convicted of murdering both her sons.
• Paediatrician Roy Meadow argued that the chance
  of both children dying naturally was 73 million
  to 1.
• Didnt take into account that double murder would
  have been more unlikely.
• Conviction overturned in 2003.

36
Exhibit 2 Denis Adams, 1996

• Convicted of rape based on DNA found at the scene
  of the crime.
• Probability of a match said to be 1 in 20
  million.
• There was no other evidence to convict victim
  did not identify Adams in a line-up and Adams had
  an alibi.
• The defence team instructed the jury in the use
  of Bayes Theorem. The judge questioned its
  appropriateness.
• After 2 appeals, Adams is still convicted.

37
A rule against Bayes

• In 2010 a convicted killer known as T appealed
  against his conviction.
• Part of the evidence was based on the special
  markings on his Nike trainers.
• The data on how many pairs of such trainers
  existed was unreliable.
• It has now been ruled that Bayes Theorem is not
  allowed in court unless the underlying statistics
  are firm.

38
Quotes of statistics

• 98 of all statistics are made up
• The average human has one breast and one
  testicle.
• Statistics are like bikinis.  What they reveal
  is suggestive, but what they conceal is vital. 
• There are three kinds of lies lies, damned
  lies, and statistics.

39
Misuse of statistics

• We are going to look at some examples of bad
  statistics in the media.
• What things should we look out for to spot bad
  maths and stats?

40
Strange patterns

• Matt Parker, of Queen Mary University London,
  look at 800 ancient sites.
• 3 sites, around Birmingham, formed a perfect
  equilateral triangle.
• Extending the base of this triangle links up 2
  more sites, more than 150 miles apart, with an
  accuracy of 0.05.

41
Ancient sites?
42
Ancient sites?
43
What to watch out for

• Events assumed to be independent (e.g. 6 double
  yolks article).
• Patterns found using large amounts of data (e.g.
  ancient sat-nav article)
• Other factors not taken into account (e.g.
  perfect whist deal article)
• Confusion of the inverse
• Omission of relevant data
• Misleading labelling of graphs

44
Lessons to take home

• Dont play the lottery.
• Think very carefully when you are asked a
  question about probability.
• Dont confuse conditional probabilities with
  their inverses.
• Ask questions whenever you see statistics in the
  media! (And write in to report bad journalism!)