MAT199: Math Alive Probability and Statistics

1. MAT199: Math Alive Probability and Statistics Ian Griffiths Mathematical Institute, University of Oxford, Department of Mathematics, Princeton University

2. Probability and Statistics • Probability arises in many forms of everyday life: e.g., • There is a one in six chance of rolling a 6 on a die.

3. Probability and Statistics • Probability arises in many forms of everyday life: e.g., • There is a one in six chance of rolling a 6 on a die. • There is a 30% chance of rain today.

4. Probability and Statistics • Probability arises in many forms of everyday life: e.g., • There is a one in six chance of rolling a 6 on a die. • There is a 30% chance of rain today. • There is a 1% chance that Princeton will win March Madness in the next ten years.

5. Disease testing • While at the doctors’ surgery, the doctor decides to do a routine test for a very rare disease which affects only 1 in 10,000 people. • The test is 99% accurate. • The test comes back positive. What should the doctor say?

6. Disease testing • While at the doctors’ surgery, the doctor decides to do a routine test for a very rare disease which affects only 1 in 10,000 people. • The test is 99% accurate. • The test comes back positive. What should the doctor say? (a) “It’s bad news I’m afraid…”

7. Disease testing • While at the doctors’ surgery, the doctor decides to do a routine test for a very rare disease which affects only 1 in 10,000 people. • The test is 99% accurate. • The test comes back positive. What should the doctor say? (a) “It’s bad news I’m afraid…” • (b) “Don’t worry, it’s still very unlikely you have the disease.”

8. Disease testing • While at the doctors’ surgery, the doctor decides to do a routine test for a very rare disease which affects only 1 in 10,000 people. • The test is 99% accurate. • The test comes back positive. What should the doctor say? (a) “It’s bad news I’m afraid…” • (b) “Don’t worry, it’s still very unlikely you have the disease.” • He should say (b). In fact, the patient still has less than 1% chance of having the disease.

9. Cups and prizes • A prize is placed under one cup. What is the probability that you guess the correct cup? 3 2 1 • The probability of guessing the correct cup is one in three.

10. Cups and prizes • A prize is placed under one cup. What is the probability that you guess the correct cup? 3 2 1 • The probability of guessing the correct cup is one in three. • You choose a cup. I then reveal that the prize is not under one of the other cups and offer you a chance to switch. Should you stick or switch? • If you stick then your chance of winning is still one in three. But what are your chances if you switch…?

11. Cups and prizes • Let’s suppose the prize is under cup 1. (The argument is exactly the same if the prize is under cup 2 or cup 3.) 3 2 1

12. Cups and prizes • Let’s suppose the prize is under cup 1. (The argument is exactly the same if the prize is under cup 2 or cup 3.) 3 2 1 • If you choose to switch then there are three possible outcomes: • 1) You choose cup 1 (containing the prize). I uncover one of the other empty cups. You switch to the remaining covered cup. You lose.

13. Cups and prizes • Let’s suppose the prize is under cup 1. (The argument is exactly the same if the prize is under cup 2 or cup 3.) 3 2 1 • If you choose to switch then there are three possible outcomes: • 1) You choose cup 1 (containing the prize). I uncover one of the other empty cups. You switch to the remaining covered cup. You lose. 2) You choose cup 2. I uncover the empty cup (cup 3). You switch to the remaining covered cup (cup 1). You win.

14. Cups and prizes • Let’s suppose the prize is under cup 1. (The argument is exactly the same if the prize is under cup 2 or cup 3.) 3 2 1 • If you choose to switch then there are three possible outcomes: • 1) You choose cup 1 (containing the prize). I uncover one of the other empty cups. You switch to the remaining covered cup. You lose. 2) You choose cup 2. I uncover the empty cup (cup 3). You switch to the remaining covered cup (cup 1). You win. • 3) You choose cup 3. I uncover the empty cup (cup 2). You switch to the remaining covered cup (cup 1). You win.

15. Cups and prizes • Let’s suppose the prize is under cup 1. (The argument is exactly the same if the prize is under cup 2 or cup 3.) 3 2 1 • If you choose to switch then there are three possible outcomes: • 1) You choose cup 1 (containing the prize). I uncover one of the other empty cups. You switch to the remaining covered cup. You lose. 2) You choose cup 2. I uncover the empty cup (cup 3). You switch to the remaining covered cup (cup 1). You win. • 3) You choose cup 3. I uncover the empty cup (cup 2). You switch to the remaining covered cup (cup 1). You win.So if you switch then you win two times out of three. This is twice as good a chance of winning than if you had stayed with your original choice.

16. Bayes’ theorem Thomas Bayes 1701–1761

17. Summary of lecture 13 • Probability of an event A occurring is denoted by p(A) • We use three rules for calculating probabilities: p(A or B) = p(A) + p(B) (1) p(A and B) = p(A) x p(B) (2) p(A givenB) = p(A and B) (3) p(B) This is called Bayes’ theorem

18. What is the probability of two people in a room sharing a birthday?

19. What is the probability of two people in a room sharing a birthday? Probability of sharing a birthday Number of people in the room

20. What is the probability of two people in a room sharing a birthday? Probability of sharing a birthday Number of people in the room • You need only 22 people in a room for the probability of two people sharing a birthday to be greater than 0.5.

21. Statistics • “I can prove anything with statistics except the truth.” George Canning.

22. Statistics • “I can prove anything with statistics except the truth.” George Canning. • Unjustified accuracy • 87.452522% of all statistics claim a precision of results that are not backed up by the method used.

23. Statistics • “I can prove anything with statistics except the truth.” George Canning. • Unjustified accuracy • 87.452522% of all statistics claim a precision of results that are not backed up by the method used. • Manipulating averages • A statistician died crossing a river that was, on average, six inches deep.

24. Statistics • “I can prove anything with statistics except the truth.” George Canning. • Unjustified accuracy • 87.452522% of all statistics claim a precision of results that are not backed up by the method used. • Manipulating averages • A statistician died crossing a river that was, on average, six inches deep. • Plain wrong • Math illiteracy affects 8 out of every 5 people. • If there is a 50/50 chance of something going wrong then 9 times out of 10 it will.

25. Summary of lecture 14 • The expected value for an event, E = (probability of event 1 happening) x (value for event 1) + (probability of event 2 happening) x (value for event 2) • +… + (probability of event N happening) x (value for event N) • The variance of an event, Variance = (probability of event 1 happening) x (value for event 1 – Expectation value, E)2+ (probability of event 2 happening) x (value for event 2 – Expectation value, E) 2+… + (probability of event N happening) x (value for event N – Expectation value, E) 2 • The standard deviation of an event = variance

26. Pascal’s triangle 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 11 6 15 20 15 6 11 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 11 9 36 84 126 126 84 36 9 11 10 45 120 210 252 210 120 45 10 1…

27. What happens if we take more points? n =10

28. What happens if we take more points? n =10 n =20

29. What happens if we take more points? n =10 n =20 n =100

30. What happens if we take more points? n =10 n =20 n =100 This is called the binomial distribution

31. Analysing the final curve • We can scale the distribution so that the sum of all the possible outcomes = 1 and the values correspond to probabilities.

32. Analysing the final curve p n = 100 r • We can scale the distribution so that the sum of all the possible outcomes = 1 and the values correspond to probabilities. • The result is approximated by the formula:

33. Analysing the final curve p n = 100 r • We can scale the distribution so that the sum of all the possible outcomes = 1 and the values correspond to probabilities. • The result is approximated by the formula: This is the mean value

34. Analysing the final curve p n = 100 r • We can scale the distribution so that the sum of all the possible outcomes = 1 and the values correspond to probabilities. • The result is approximated by the formula: This is the mean value This is related to the spread of the data

35. The normal distribution

36. The normal distribution • In general we write:

37. The normal distribution • In general we write: The mean value The standard deviation

38. The normal distribution • In general we write: The mean value The standard deviation • This is called the normal distribution.

39. The normal distribution • The normal distribution is also known as a bell curve.

40. Summary of lecture 15 • The normal distribution: Tossing a coin 100 times. This shows the probability, p, of getting a head a total of r times. p r

41. Summary of lecture 15 • The normal distribution: Tossing a coin 100 times. This shows the probability, p, of getting a head a total of r times. p r

42. Summary of lecture 15 • The normal distribution: Tossing a coin 100 times. This shows the probability, p, of getting a head a total of r times. In this example the mean is m = 50and the standard deviation is σ = 10. p The mean value r The standard deviation

43. Summary of lecture 15 • The normal distribution: Tossing a coin 100 times. This shows the probability, p, of getting a head a total of r times. In this example the mean is m = 50and the standard deviation is σ = 10. p The mean value r The standard deviation • This kind of distribution arises in many situations, e.g. distribution of heights of people in the U.S.

44. The Griffiths University

45. The Griffiths University • Here are some statistics for the 2012 intake of students at the Griffiths University:Success rate for males who applied to the Griffiths University: 17%Success rate for females who applied to the Griffiths University: 82%

46. The Griffiths University • Here are some statistics for the 2012 intake of students at the Griffiths University:Success rate for males who applied to the Griffiths University: 17%Success rate for females who applied to the Griffiths University: 82% • Is the Griffiths University a fair University? (a) Yes.(b) No.(c) Don’t know.

47. Representing data Good forms of data representation

48. Representing data Good forms of data representation • This is a visualization of the military campaign of Bonaparte against Russia. The band shows the route taken by Napoleon’s army. The thickness of the band is proportional to the number of surviving troops.

49. Representing data Good forms of data representation

50. Representing data Good forms of data representation • This shows the air pollution in Los Angeles at different times of the day.