David Spiegelhalter

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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

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My background

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Whats 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

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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
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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
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Activity when does the first head come up?

• All stand up
• Toss a coin together
• Sit down when you get a head

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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.

Name 1 2 3 4
Abby 4 2 2 1
Ben 1 2 1 2

Totals 5 4 3 3
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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
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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)
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• What would be the longest wait recorded in 1000
  trials?

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• Buying a lottery ticket but which number to
  choose?
• 6/49 lottery

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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 !!

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Some statistics
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Counts obey the rules of probability
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What is the distribution of gaps?As expected, a
geometric distributionBut is a maximum gap
length of 72 surprising?
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Simulate 1000 full lottery histories72 is
almost exactly the expected maximum gap
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• How much of the English Premier football league
  is due to chance?

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• Coincidences
• three children born on the same day?

• 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?

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• Coincidences

• Whats 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

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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

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• Coincidences

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

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Activity coincidences

• Birthdays
• In your class, do any of you have the same
  birthday?

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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

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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

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Activity choosing numbers

• Choose a number between 1 and 100, but dont say
  what it is to anyone.
• When your number is called out, stand up.

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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

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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

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Pick the same number?

• Assume N people each choose a whole numbers
  between 1 and T
• Each pair 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

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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!