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 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?

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

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… 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 prioriodds – 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(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.

30. Importance of Bayes’ Theorem • Bayes’ Theorem is especially useful in medicine and in law. • Most doctors get the following question wrong. Let’s 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 don’t 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 don’t have breast cancer. • 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%.

33. The prosecutor’s 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 prosecutor’s fallacy • 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.

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. • Didn’t 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 • 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!)