Title: Mathematical Ideas that Shaped the World
1Mathematical Ideas that Shaped the World
2Plan 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?
3An unfortunate truth
- Humans have an extraordinarily bad intuition
about probability.
4Winning 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
5Is 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.
6Is winning the lottery more or less likely?
LESS
MORE
1 in 24,000
- Dying from a road accident in 1 year
7Is winning the lottery more or less likely?
LESS
MORE
1 in 25 million
- Dying in the next flight you take
8Is winning the lottery more or less likely?
LESS
MORE
1 in 1 million
- Being struck by lightning
9Is winning the lottery more or less likely?
LESS
MORE
1 in 300 million
- Dying from a shark attack
10Is winning the lottery more or less likely?
LESS
MORE
1 in 2 million
- Dying in the next hour from any causes whatsoever
11Conclusion
- 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.
12The 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
13Number 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
14The birthday graph
15In 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
16Monty 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?
17Suppose 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.
18Conditional 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?
19How 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.
20Boy 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
21Explanation
- 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.
22A 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?!
23Homework
- 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?
24Confusion 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.
25Manipulating 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?
26Bayesian 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.
27Thomas 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.
28Thomas 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.
29Bayes 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.
30Importance 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!
31A 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?
32Doing 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.
33The 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?
34The 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.
35Exhibit 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.
36Exhibit 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.
37A 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.
38Quotes 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.
39Misuse 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?
40Strange 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.
41Ancient sites?
42Ancient sites?
43What 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
44Lessons 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!)