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What are my odds?

What are my odds?. 13 th June 2011 Dr Andrew Duncan. What is your opinion of maths?. Boring and useless. Boring but useful. Interesting but useless. Interesting and useful. Uses of maths and stats. Fighting crime …well, solving crime. The stock market (making money).

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What are my odds?

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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…

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