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. Tuesday, February 12, 13

Flip a fair coin twice. If you get TT, re-roll. What’s the probability of HH? Tuesday, February 12, 13

HH HT TH TT Tuesday, February 12, 13

HH HT TH TT Tuesday, February 12, 13

HH HT prob = 1/3 TH TT Tuesday, February 12, 13

0.25 0.25 HH 0.25 0.25 HT 0.25 0.25 TH 0.25 0.25 TT Tuesday, February 12, 13

0.25 0.25 HH 0.25 0.25 HT 0.25 0.25 TH 0.25 0.25 TT Tuesday, February 12, 13

0.25 0.25 0.333 HH 0.25 0.25 0.333 HT 0.25 0.25 0.333 TH 0.25 0.25 TT Tuesday, February 12, 13

0.25 0.25 0.333 HH 0.25 0.25 0.333 HT 0.25 0.25 0.333 TH 0.25 0.25 TT prob = 0.333 Tuesday, February 12, 13

A = all heads G = not all tails What is P(A | G)? Tuesday, February 12, 13

What if I flip the coin 3 times? (Again, re-rolling whenever I get all tails.) Tuesday, February 12, 13

A = all heads G = not all tails What is P(A | G)? Solution (in this case): P(A) / P(G) Tuesday, February 12, 13

The general rule: P(A | B) = P(A and B) / P(B) Tuesday, February 12, 13

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? Tuesday, February 12, 13

all Tuesday, February 12, 13

all Tuesday, February 12, 13

live to 60 all Tuesday, February 12, 13

live to 60 all Tuesday, February 12, 13

live to 80 live to 60 all Tuesday, February 12, 13

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? Tuesday, February 12, 13

S = live to 60 E = live to 80 Tuesday, February 12, 13

S = live to 60 E = live to 80 P(E | S) = ? Tuesday, February 12, 13

S = live to 60 E = live to 80 P(E | S) = P(E and S) / P(S) Tuesday, February 12, 13

S = live to 60 E = live to 80 P(E | S) = P(E and S) / P(S) = 0.57 / 0.90 Tuesday, February 12, 13

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? Tuesday, February 12, 13

Out of 1000 girls: • ? die before 60 • ? live to 80 or higher • ? die between 60 and 80 Tuesday, February 12, 13

Out of 1000 girls: • 100 die before 60 • 570 live to 80 or higher • 330 die between 60 and 80 Tuesday, February 12, 13

Out of 1000 girls: • 100 die before 60 • 570 live to 80 or higher • 330 die between 60 and 80 Tuesday, February 12, 13

Out of 1000 girls: • 100 die before 60 • 570 live to 80 or higher • 330 die between 60 and 80 Tuesday, February 12, 13

Out of 1000 girls: • 100 die before 60 • 570 live to 80 or higher • 330 die between 60 and 80 prob = 570/(330+570) Tuesday, February 12, 13

conditional probability: eliminate the impossible Tuesday, February 12, 13

Tuesday, February 12, 13

Tuesday, February 12, 13

In 1978, the NY Times reported that the occupation that had the longest life expectancy was.... Tuesday, February 12, 13

In 1978, the NY Times reported that the occupation that had the longest life expectancy was.... orchestra conductor. Tuesday, February 12, 13

In 1978, the NY Times reported that the occupation that had the longest life expectancy was.... orchestra conductor. Should you take up conducting? Tuesday, February 12, 13

P(live to 80 | live to 55 and conductor) P(live to 80) Tuesday, February 12, 13

P(live to 80 | live to 55 and conductor) P(live to 80 | live to 55) Tuesday, February 12, 13

P(live to 80 | conductor) P(live to 80 | affluent) Tuesday, February 12, 13

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? Tuesday, February 12, 13

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! Tuesday, February 12, 13

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? Tuesday, February 12, 13

Doctors’ estimates ranged from 0% - 100% Tuesday, February 12, 13

Method 1: shut up and calculate Tuesday, February 12, 13

Method 2: natural frequencies Tuesday, February 12, 13

Tuesday, February 12, 13

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. Tuesday, February 12, 13