Mathematical Ideas that
Shaped the World
Bayesian Statistics
Plan for this class
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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 someone’s DNA is found at a crime scene,
what is the chance they are guilty?
How can we spot bad statistics in the media?
An unfortunate truth
Humans have an
extraordinarily bad
intuition about
probability.
Winning the lottery
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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…
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.
Is winning the lottery more or less likely?
LESS
MORE
1 in 24,000
Dying from a road accident in 1 year
Is winning the lottery more or less likely?
LESS
MORE
1 in 25 million
Dying in the next flight you take
Is winning the lottery more or less likely?
LESS
MORE
1 in 1 million
Being struck by lightning
Is winning the lottery more or less likely?
LESS
MORE
1 in 300 million
Dying from a shark attack
Is winning the lottery more or less likely?
LESS
MORE
1 in 2 million
Dying in the next hour from any causes
whatsoever
Conclusion
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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.
The birthday problem
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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
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%
The birthday graph
In this room?
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What is the chance that two people in this
room have birthdays less than 3 days apart
(ignoring the year?)
Answer: more than 50%
Monty Hall
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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?
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.
Conditional probability
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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?
How to think about conditional
probability
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Conditional probability is all about updating
your odds in light of new evidence.
There are a priori odds – the initial probability
of an event.
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E.g. the probability of rolling a 6 is a priori 1 in 6.
After new evidence, you have a posteriori
odds.
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E.g. the probability of having a 6, given that the
sum of two dice is 8, is 1 in 5.
Boy or girl?
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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
Explanation
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A priori, there are 4 possible combinations of
children:
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Boy – Boy
Boy – Girl
Girl – Boy
Girl - Girl
From our new evidence, we know that GirlGirl is not possible, leaving only 3 options.
Of these 3 options, only one of them is BoyBoy.
A paradox?
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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?!
Homework
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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?
Confusion of the inverse
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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.
Manipulating statistics
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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?”
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But what proportion of illiterate people were
criminals?
Bayesian statistics
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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.
Thomas Bayes (1702 – 1761)
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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.
Thomas Bayes (1702 – 1761)
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During his lifetime, Bayes only published two
papers.
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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”.
Bayes’ Theorem
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P(A) is the prior probability of A.
P(B) is the prior probability of B.
P(A|B) is the probability of A happening,
given that B has happened.
P(B|A) is the probability of B happening, given
that A has happened.
Importance of Bayes’ Theorem
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Bayes’ Theorem is especially useful in
medicine and in law.
Most doctors get the following question
wrong. Let’s see what you think!
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 don’t have breast
cancer, 10% of them will get a positive result.
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If a woman tests positive for breast cancer,
what is the chance she has actually has it?
Doing the numbers
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Consider 10,000 women.
100 of them will have breast cancer.
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9900 of them don’t have breast cancer.
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80 of them test positive
20 of them test negative
990 of them test positive
8910 of them test negative
In total there are (80+990) = 1070 positive
results, of which only 80 have cancer.
That’s 7.4%.
The prosecutor’s fallacy
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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?
The prosecutor’s fallacy
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If the a priori chance of the defendant’s 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.
Exhibit 1: Sally Clark, 1999
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Convicted of murdering both
her sons.
Paediatrician Roy Meadow
argued that the chance of both
children dying naturally was 73
million to 1.
Didn’t take into account that
double murder would have
been more unlikely.
Conviction overturned in 2003.
Exhibit 2: Denis Adams, 1996
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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.
A rule against Bayes
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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”.
Quotes of statistics
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“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.“
Misuse of statistics
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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?
Strange patterns
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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%.
Ancient sites?
Ancient sites?
What to watch out for
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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
Lessons to take home
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Don’t play the lottery.
Think very carefully when you are asked a
question about probability.
Don’t confuse conditional probabilities with
their inverses.
Ask questions whenever you see statistics in
the media! (And write in to report bad
journalism!)