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  1. What are my odds? 13th June 2011 Dr Andrew Duncan

  2. What is your opinion of maths? • Boring and useless. • Boring but useful. • Interesting but useless. • Interesting and useful.

  3. Uses of maths and stats • Fighting crime • …well, solving crime. • The stock market (making money). • Predicting the spread of disease. • Games of chance.

  4. What are my odds? • Probability and a little statistics. • Deal or No Deal • The Monty Hall Problem • The National Lottery • Numbers in the press • Birthdays

  5. 1p 10p 50p £1 £5 £10 £50 £100 £250 £500 £750 £1,000 £3,000 £5,000 £10,000 £15,000 £20,000 £35,000 £50,000 £75,000 £100,000 £250,000 Deal or No Deal

  6. Deal or No Deal • £1, £500, £35000, £50000 • Banker offers £26000 • Deal or No Deal?

  7. £1, £500, £35000, £50000.Banker offers £26000. • Deal. • No Deal.

  8. Deal or No Deal • £1, £500, £35000, £50000 • Banker offers £26000 • Deal or No Deal • Estimate expected winnings • Same as average – equal chance. • Average ≈ £21,000. • Deal or No Deal?

  9. Deal or No Deal • £3000, £10000, £15000, £100000. • Banker offers £26000. • Deal or No Deal?

  10. £3000, £10000, £15000, £100000.Banker offers £26000. • Deal. • No Deal.

  11. Deal or No Deal • £3000, £10000, £15000, £100000 • Banker offers £26000 • Deal or No Deal? • Average = £32,000 • Deal or No Deal? • Average of original 22 boxes = £25,712.19 • Obviously a little different when on TV

  12. The Monty Hall Problem • Pick a door…

  13. The Monty Hall Problem • You’ve picked your door. • The host (Monty) reveals a banana. • Then he offers you a choice…

  14. Do you want to switch doors? • Don’t be daft, no thanks. • Yes please.

  15. The Monty Hall Problem • You should switch – always. Why? • Assume we start with door #1. • Win 2 out of 3 if we switch. But not always.

  16. The Monty Hall Problem An alternative explanation 1/3 2/3

  17. The Monty Hall Problem Then Monty reveals the banana 1/3 2/3 1/3 2/3 0

  18. It’s a Lottery • Millions play the National Lottery each week. • Are you going to win the jackpot? • Lotto • select 6 from 49. • 1 in 14 million chance of winning Lotto jackpot. Why?

  19. It’s a Lottery • Odds of getting all six numbers correct? • First number drawn matches one on ticket – • Second number drawn matches one on ticket – • All six?

  20. WARNING! WARNING! Be Aware. Complicated-looking mathematical formula approaching. Please ensure your seat backs and tray tables are in the upright position. Turn off all electrical equipment.

  21. nCr • Alternative method. • n things – Choose – r of them. Order doesn’t matter, but can’t repeat.

  22. nCr • Alternative method. • 49 numbers – Choose – 6 of them. Order doesn’t matter, but can’t repeat.

  23. nCr • ! – factorial. • 6! = 6x5x4x3x2x1 and • 49!=49x48x47x46x45x44x43x42x…x3x2x1 =49x48x47x46x45x44x43!

  24. Combinations • So 49C6gives 13,983,816 combinations. • Chance of winning the jackpot is 1 in 13,983,816. • 5 card hands in poker - 52C5 – 2,598,960. • 3 heads from 7 coin flips - 7C3 – 35.

  25. Combinations • For small numbers it is easier to use Pascal’s Triangle. • 7C3=35

  26. Pascal’s Triangle • Fibonnaci’s sequence – 0,1,1,2,3,5,8,13,21,… 0 1 1 2 3 5 8 13 21

  27. Pascal’s Triangle • Or the square numbers – 1,4,9,16,25,36,…

  28. Sierpinski’s Triangle • Fractal shape. Source: http://www.zeuscat.com/andrew/chaos/sierpinski.html

  29. Sierpinski’s Triangle • End up with Source: http://www.zeuscat.com/andrew/chaos/sierpinski.html

  30. Sierpinski’s Triangle

  31. Sierpinski’s Triangle

  32. Sierpinski’s Triangle

  33. Sierpinski’s Triangle • Compare with Source: http://www.zeuscat.com/andrew/chaos/sierpinski.html

  34. Headline News • World Cancer Research Fund 2007 “Avoid processed meat.” “…extra ounce of bacon a day increased risk of colorectal cancer by 21%.” • BBC News November 2002 “For every alcoholic drink a woman consumes, her risk of breast cancer rises 6%.” • News Headlines January 2005 Mobile phones double risk of acoustic neuroma.

  35. Making a pig’s ear of it • “…extra ounce of bacon a day increased risk of colorectal cancer by 21%.” • What does 21% increase mean? • Begin with 10% increase…

  36. If something increases by 10% it • goes up by 10 • goes up by 100 • goes up by 1 • goes up by 0.1 • None of 1,2,3,4 • All of 1,2,3,4

  37. Making a pig’s ear of it • Any % increase is relative. • 10% increase means 10% more than you had to start with. • £1 → £1.10, 10p increase. • £100 → £110, £10 increase. • £10000 → £11000, £1000 increase. • All 10%, all relative.

  38. Making a pig’s ear of it • 21% increase in colorectal cancer. • Work with natural frequencies. • base rate of 5 in 100. • 21% of 5 ≈ 1. • new rate – 6 in 100. • Extra ounce bacon – 1 more person in every 100.

  39. mmm Beer… • BBC News November 2002 “For every alcoholic drink a woman consumes, her risk of breast cancer rises 6%.” • Actually… Cancer Research UK 2002 “A woman’s risk of breast cancer increases by 6% for every extra alcoholic drink consumed on a daily base, the world’s largest study of women’s smoking and drinking behaviour reveals.” • An extra drink a day increases risk by 6%.

  40. mmm Beer… • 6% of what? • base rate of 9 in 100 (or 18 in 200). • 6% of 18 ≈ 1. • new rate – 19 in 200. • Extra drink a day means 1 more in every 200.

  41. I’m on the phone… • Mobile phones double risk of acoustic neuroma – 100% increase. • Double (or 100%) of what? • base rate of 1 in 100,000. • new rate – 2 in 100,000.

  42. Presentation of Percentages • Not saying the %’s are wrong. • Read further – find the base rates. • Work out the new rates.

  43. Happy birthday to you … and you … and you … and you … • How many people do you need in a room to guarantee two or more of them have the same birthday? • 366 (normal year, not leap). • What are the chances with less than 366? • How about with 23?

  44. What are the chances that with 23 people in a room, two or more of them share a birthday? • 1% chance (1 in 100) • 5% chance (1 in 20) • 10% chance (1 in 10) • 25% chance (1 in 4) • 50% chance (1 in 2)

  45. Happy birthday to you … and you … and you … and you … • 23 people means slightly better than 50% chance two or more share a birthday. • Why? • Easier to look at it backwards - chances no one has the same birthday. • Start with 1 person.

  46. Happy birthday to you … and you … and you … and you … • 2 people? • 3 people?

  47. Happy birthday to you … and you … and you … and you … • And for 23 people • For 23 people probability of 49.3% that none of them share a birthday.

  48. Happy birthday to you … and you … and you … and you … • That means probability that two or more share birthday is or 50.7%.

  49. Happy birthday to you … and you … and you … and you … • How about some other groups of people

  50. Happy birthday to you … and you … and you … and you … • This morning there were 67 tickets booked for this talk. So we have a 99.8% chance of two or more people in the room sharing a birthday. • You were asked to tick your birthday on a calendar when you came in. Well…