2. Conditional Probability and Independence

2. Conditional Probability and Independence

Conditional Probability

• : prior, : posterior

Elder is a girl vs. At least one girl vs. Random child is a girl

• at least one is a girl:

• P(both are girls | at least one is a girl):

• P(both are girls | elder is a girl):

• P(both are girls | already see that one is a girl):
• older/younger(1/2) - ramdomly see girl(1/2) are independence

A girl born in winter

• P(at least one winter girl):
• P(winter girl):

• P(both are girls & at least one winter girl):
• P(both are girls) x P(at least one winter child)

• P(both are girls | at least one winter girl)

Bayes’ Rule

LOTP(Law of total probability)

Random coin

• one fair coin, one biased coin(H: 3/4)

• : head 3 times, : pick fair coin

Testing for rare disease

• : sensitivity, specificity

Bayes’ Rule with extra conditioning

Random coin 2

• one fair coin, one biased coin(H: 3/4)

• : head 3 times, : head on 4th time, : fair coin

Independence

• Pairwise independence doesn’t imply independence

Conditional Independence

• Conditional independence doesn’t imply independence

• Independence doesn’t imply conditional independence

• Conditional independence given E vs. E^c are different

Example

Testing for rare disease 2

Monty Hall problem

Branching process

• , ,

Prosecutor’s fallacy

• : two sons sudden death

• : she is guility ↔ innocent

• 피고 범인 → 확실한 증거가 아니라 확실한 증거(조건) → 피고 범인 이여야 함

Defense attorney’s fallacy

• : guility, : abuse, : murder

Simson’s paradox