David Spiegelhalter Winton Professor of the Public Understanding of Risk, Statistical Laboratory, University of Cambrid

1. Why do unlikely things happen so often? David Spiegelhalter Winton Professor of the Public Understanding of Risk, Statistical Laboratory, University of Cambridge Motivate, October 8th 2008

2. My background

3. What’s this all about? • Probability and how it plays out in the real world • Looking at the pattern of ‘waiting times’ • Working out the chances of specific unlikely events • Being able to see whether an apparently rare event is really surprising

5. Probability from ‘symmetry’ Number of ways of ‘winning’ Number of equally likely outcome e.g. throw dice, probability of a ‘six’ = 1/6 Or 0.1666 , or 16.7%

6. Rules of probability ‘OR’ means you ADD ‘AND’ means you MULTIPLY 1 dice: Prob(1 or 2) = 1/6 + 1/6 = 1/3 2 dice: Prob(6 and 6) = 1/6 x 1/6 = 1/36

7. Activity: when does the first head come up? • All stand up • Toss a coin together • Sit down when you get a head

8. Activity: when does the first head come up? • Everyone does 10 runs. • A run means tossing a coin until you get a head. • Record the number of coin tosses, INCLUDING the time you got the head, so for T T T H count 1 in the 4 column. • Each person adds their data to the class sheet. • Report totals for whole class when asked.

9. When does the first head come up? Prob(1st is head)= 1/2 Prob(2nd is the first head) = Prob(1st is tail and 2nd is head) = 1/2 x 1/2 = 1/4 Prob(3rd is the first head) = Prob(1st is tail and 2nd is tail and 3rd is head) = = 1/2 x 1/2 x 1/2 = 1/8 Prob(nth is the first head) = 1/2n

10. A geometric distribution - probability of first head on nth toss (This is also the probability of having to wait more than n tosses until the first head)

11. What would be the longest wait recorded in 1000 trials?

13. Buying a lottery ticket – but which number to choose? • “6/49” lottery

14. What is the chance of winning? Imagine that the numbers on your lottery ticket were labelled as WIN Chance of picking first WIN ball = 6/49 Chance of picking second WIN = 5/48 Chance of picking all WIN balls = 6/49 x 5/48 x 4/47 x 3/46 x 2/45 x 1/44 = = 1/13,983,816 !!

15. Some statistics…

16. Counts obey the rules of probability

17. What is the distribution of gaps?As expected, a geometric distributionBut is a maximum gap length of 72 surprising?

18. Simulate 1000 full lottery histories72 is almost exactly the expected maximum gap

20. How much of the English Premier football league is due to chance?

22. The MacKriell family in Gloucester: Robin 14, Rebecca 12, Ruby 0, all born on January 29th • 1/365 x 1/365 chance (assuming uniform birth-dates) = 7.5 in 1,000,000 • But there are 1,000,000 families in the UK with 3 children • So where are the other examples? Coincidences – three children born on the same day?

23. What’s the chance p of the specific event? • How many opportunities N are there for a ‘similar’ event to occur? • Multiply to give expected number E = Np Coincidences –

24. So why does anyone win the lottery? • Each ticket has around 1/14,000,000 chance of winning • They sell around 30,000,000 tickets • So the expected number of Jackpot winners is around 2 • So the chance that nobody wins (a rollover) is around 0.13

25. Joyce and Ron Pulsford of Pagham near Bognor Regis were both 80 on 08.08.08 • Are they unique in the country? Coincidences –

26. Activity: coincidences Birthdays: • In your class, do any of you have the same birthday?

27. Coincidences Birthdays: • 23 people: 51% chance that 2 share a birthday • 35 people: 81% chance that 2 share a birthday • 80 people: 99.99% chance that 2 share a birthday

28. Why does this happen? • Imagine 35 people in a line • First birthday can be anything • 2nd birthday must be different from first: probability 364/365 = 0.997 • 3rd birthday must be different from 1st and 2nd: probability 363/365 = 0.995 • ….. • Probability that all 35 are different = 0.997 x 0.995 x …. 0.907 = 0.19

29. Activity: choosing numbers • Choose a number between 1 and 100, but don’t say what it is to anyone. • When your number is called out, stand up.

30. 20 people choosing numbers between 1 and 100 • First number can be anything • 2nd number must be different from first: probability 99/100 = 0.99 • 3rd number must be different from 1st and 2nd: probability 98/100 = 0.98 • ….. • Probability that all 20 are different = 0.99 x 0.98 x …. 0.81 = 0.13

31. A neat trick • Assume N people each choose a number between 1 and T • Set T = (N/2)2 • eg N = 20; T = 100 N = 40; T = 400 N =400; T = 40000 • Then the probability that all choose different numbers  0.13

32. Pick the same number? • Assume N people each choose a whole numbers between 1 and T • Eachpair has a 1/T chance of matching • N(N-1)/2  N2/2 pairs of people • So expected matches E  N2/(2T) • So if T = (N/2)2 , then E  2 • So Prob “all different”  0.13

33. Summary • Probability really works in the world • Can predict patterns that ‘random’ events make, even if cannot predict individual events • Coincidences happen because there are many opportunities for rare events • Maths can help!