Conditional probability

Transcription

1 Conditional probability February 12, 2012

2 Once you eliminate the impossible, whatever remains, however improbable, must be the truth.

3 Flip a fair coin twice. If you get TT, re-roll.

4 Flip a fair coin twice. If you get TT, re-roll. What s the probability of HH?

5 HH HT TH TT

6 HH HT TH TT

7 HH HT TH prob = 1/3 TT

8 HH HT TH TT

9 HH HT TH TT

10 HH HT TH TT

11 HH HT TH TT prob = 0.333

12 A = all heads G = not all tails What is P(A G)?

13 What if I flip the coin 3 times? (Again, re-rolling whenever I get all tails.)

14 A = all heads G = not all tails What is P(A G)? Solution (in this case): P(A) / P(G)

15 The general rule: P(A B) = P(A and B) / P(B)

16 90% of girls born today will live to age 60, while only 57% will live to age to 80. If a girl born today lives to age 60, what s the probability she lives to 80?

17 all

18 all

19 live to 60 all

20 live to 60 all

21 live to 80 live to 60 all

22 90% of girls born today will live to age 60, while only 57% will live to age to 80. If a girl born today lives to age 60, what s the probability she lives to 80?

23 S = live to 60 E = live to 80

24 S = live to 60 E = live to 80 P(E S) =?

25 S = live to 60 E = live to 80 P(E S) = P(E and S) / P(S)

26 S = live to 60 E = live to 80 P(E S) = P(E and S) / P(S) = 0.57 / 0.90

27 90% of girls born today will live to age 60, while only 57% will live to age to 80. If a girl born today lives to age 60, what s the probability she lives to 80?

28 Out of 1000 girls:? die before 60? live to 80 or higher? die between 60 and 80

29 Out of 1000 girls: 100 die before live to 80 or higher 330 die between 60 and 80

30 Out of 1000 girls: 100 die before live to 80 or higher 330 die between 60 and 80

31 Out of 1000 girls: 100 die before live to 80 or higher 330 die between 60 and 80

32 Out of 1000 girls: 100 die before live to 80 or higher 330 die between 60 and 80 prob = 570/( )

33 conditional probability: eliminate the impossible

34

35

36 In 1978, the NY Times reported that the occupation that had the longest life expectancy was...

37 In 1978, the NY Times reported that the occupation that had the longest life expectancy was... orchestra conductor.

38 In 1978, the NY Times reported that the occupation that had the longest life expectancy was... orchestra conductor. Should you take up conducting?

39 P(live to 80 live to 55 and conductor) P(live to 80)

40 P(live to 80 live to 55 and conductor) P(live to 80 live to 55)

41 P(live to 80 conductor) P(live to 80 affluent)

42 If a woman has breast cancer, there s a 90% probability she will have a positive mammogram. If a woman doesn t, there s a 7% probability she will still have a positive mammogram. What is the probability a randomly selected woman who has a positive mammogram actually has breast cancer?

43 If a woman has breast cancer, there s a 90% probability she will have a positive mammogram. If a woman doesn t, there s a 7% probability she will still have a positive mammogram. What is the probability a randomly selected woman who has a positive mammogram actually has breast cancer? Not enough information!

44 If a woman has breast cancer, there s a 90% probability she will have a positive mammogram. If a woman doesn t, there s a 7% probability she will still have a positive mammogram. 0.8% of women have breast cancer. What is the probability a randomly selected woman who has a positive mammogram actually has breast cancer?

45 Doctors estimates ranged from 0% - 100%

46 Method 1: shut up and calculate

47 Method 2: natural frequencies

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49 Be careful: what does that 0.8% even mean? Where did it come from? Who does it include? Be careful: lifetime risk of getting breast cancer is much larger than 0.8%. These are the kind of questions to ask, when reading about these sorts of analyses.