SLIDE 1
Conditional probability
February 12, 2012
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“Once you eliminate the impossible, whatever remains, however improbable, must be the truth.”
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Conditional probability February 12, 2012 Tuesday, February 12, 13 - - PowerPoint PPT Presentation
Conditional probability February 12, 2012 Tuesday, February 12, 13 - - PowerPoint PPT Presentation
Conditional probability February 12, 2012 Tuesday, February 12, 13 Once you eliminate the impossible, whatever remains, however improbable, must be the truth. Tuesday, February 12, 13 Flip a fair coin twice. If you get TT, re-roll.
SLIDE 2
SLIDE 3
Flip a fair coin twice. If you get TT, re-roll.
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SLIDE 4
Flip a fair coin twice. If you get TT, re-roll. What’s the probability of HH?
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SLIDE 5
HH HT TH TT
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SLIDE 6
HH HT TH TT
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SLIDE 7
HH HT TH TT prob = 1/3
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SLIDE 8
HH HT TH TT 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
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SLIDE 9
HH HT TH TT 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
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SLIDE 10
HH HT TH TT 0.25 0.25 0.25 0.25 0.333 0.333 0.333 0.25 0.25 0.25 0.25
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SLIDE 11
HH HT TH TT 0.25 0.25 0.25 0.25 0.333 0.333 0.333 prob = 0.333 0.25 0.25 0.25 0.25
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SLIDE 12
A = all heads G = not all tails What is P(A | G)?
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SLIDE 13
What if I flip the coin 3 times? (Again, re-rolling whenever I get all tails.)
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SLIDE 14
A = all heads G = not all tails What is P(A | G)? Solution (in this case): P(A) / P(G)
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SLIDE 15
The general rule: P(A | B) = P(A and B) / P(B)
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SLIDE 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?
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SLIDE 17
all
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SLIDE 18
all
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SLIDE 19
all live to 60
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SLIDE 20
all live to 60
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SLIDE 21
all live to 60 live to 80
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SLIDE 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?
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SLIDE 23
S = live to 60 E = live to 80
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SLIDE 24
S = live to 60 E = live to 80 P(E | S) = ?
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SLIDE 25
S = live to 60 E = live to 80 P(E | S) = P(E and S) / P(S)
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SLIDE 26
S = live to 60 E = live to 80 P(E | S) = P(E and S) / P(S) = 0.57 / 0.90
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SLIDE 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?
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SLIDE 28
Out of 1000 girls:
- ? die before 60
- ? live to 80 or higher
- ? die between 60 and 80
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SLIDE 29
Out of 1000 girls:
- 100 die before 60
- 570 live to 80 or higher
- 330 die between 60 and 80
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SLIDE 30
Out of 1000 girls:
- 100 die before 60
- 570 live to 80 or higher
- 330 die between 60 and 80
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SLIDE 31
Out of 1000 girls:
- 100 die before 60
- 570 live to 80 or higher
- 330 die between 60 and 80
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SLIDE 32
Out of 1000 girls:
- 100 die before 60
- 570 live to 80 or higher
- 330 die between 60 and 80
prob = 570/(330+570)
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SLIDE 33
conditional probability: eliminate the impossible
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SLIDE 34
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SLIDE 35
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SLIDE 36
In 1978, the NY Times reported that the
- ccupation that had the longest life
expectancy was....
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SLIDE 37
In 1978, the NY Times reported that the
- ccupation that had the longest life
expectancy was.... orchestra conductor.
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SLIDE 38
In 1978, the NY Times reported that the
- ccupation that had the longest life
expectancy was.... orchestra conductor. Should you take up conducting?
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SLIDE 39
P(live to 80 | live to 55 and conductor) P(live to 80)
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SLIDE 40
P(live to 80 | live to 55 and conductor) P(live to 80 | live to 55)
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SLIDE 41
P(live to 80 | conductor) P(live to 80 | affluent)
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SLIDE 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?
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SLIDE 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!
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SLIDE 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?
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SLIDE 45
Doctors’ estimates ranged from 0% - 100%
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SLIDE 46
Method 1: shut up and calculate
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SLIDE 47
Method 2: natural frequencies
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SLIDE 48
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SLIDE 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.
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