1. Proportions

2. First, some more probability! Why were there so many birthday pairs? 14 people 91 pairs

3. Permutation An ordered set of events Boy, Girl Girl, Boy

4. Combination An unordered set of events Boy, Girl Girl, Boyor

6. Permutations - Example If you have five children, three boys and two girls, how many possible birth orders are there? Example: BGBGB

8. Checking... BBBGGBBGBGBBGGBBGBBGBGBGB BGGBBGBBBGGBBGBGBGBBGGBBB Answer = 10 permutations One combination (three boys, two girls)

9. 0!=1 1!=1

10. Multiplication rule If two events A and B are independent, then Pr[A and B] = Pr[A] x Pr[B]

11. Permutations and probabilities The multiplication rule also holds for permutations As long as the events are independent!

12. Pr[BBBGG]=Pr[B]*Pr[B]*Pr[B]*Pr[G]*Pr[G] =0.512*0.512*0.512*0.488*0.488 =(0.512) 3 *(0.488) 2

13. Probability of a combination Consider the previous example: What is the probability of the combination three boys, two girls?

14. Probability of a combination = (Number of permutations giving that combination)  (Probability of one of those permutations) * Assumes that events are independent

15. Probability of a combination =10  0.031 * Assumes that events are independent = 0.31

16. Generalizing… =  p x (1-p) n-x * Assumes that events are independent

17. The probability of a combination of independent events

18. The binomial distribution

19. Binomial distribution The probability distribution for the number of “successes” in a fixed number of independent trials, when the probability of success is the same in each trial

20. The binomial distribution: P(x) = probability of a total of x successes p = probability of success in each trial n = total number of trials

21. Sheep null distribution from computer simulation

22. Binomial distribution, n = 20, p = 0.5

23. Remember:

24. Example: Paradise flycatchers

25. A population of paradise flycatchers has 80% brown males And 20% white. You capture 5 male flycatchers at random. What is the probability that 3 of these are brown and 2 are white?

26. Example: Paradise flycatchers A population of paradise flycatchers has 80% brown males And 20% white. You capture 5 male flycatchers at random. What is the probability that 3 of these are brown and 2 are white? n = 5x = 3p = 0.8

27. Example: Paradise flycatchers A population of paradise flycatchers has 80% brown males And 20% white. You capture 5 male flycatchers at random. What is the probability that 3 of these are brown and 2 are white? n = 5x = 3p = 0.8

28. Example: Paradise flycatchers A population of paradise flycatchers has 80% brown males And 20% white. You capture 5 male flycatchers at random. What is the probability that 3 of these are brown and 2 are white? n = 5x = 3p = 0.8

30. Mean: Variance: Properties of the binomial distribution

32. Proportion of successes in a sample

33. Example: A coin is flipped 8 times; 5 were heads Proportion of heads = 5/8

34. Properties of proportions Mean = p Variance = Standard error =

35. Sampling distribution of a proportion

36. The law of large numbers The greater the sample size, the closer an estimate of a proportion is likely to be to its true value. Sample size

37. Estimating a proportion

38. Confidence interval for a proportion* where Z = 1.96 for a 95% confidence interval * The Agresti-Couli method

39. Example: The daughters of radiologists 30 out of 87 offspring of radiologists are males, and the rest female. What is the best estimate of the proportion of sons among radiologists?

40. Hypothesis testing on proportions The binomial test

41. Binomial test

42. Example

43. Hypotheses

44. N =10, p 0 =0.2

45. P =0.12 The is greater than the common  value of 0.05, so we would not reject the null hypothesis.

46. Summary