Conditional Probability:

$P(A|B)=\frac{P(A\cap B)}{P(B)}$

• $P(A\cap B)=P(B).P(A|B)=P(A).P(B|A)$:
  • Probability of the intersection of 2 events
• $P(A_1,A_2,A_3,...,A_n)=P(A_1).P(A_2|A_1).P(A_3|A_1,A_2)...P(A_n,A_1,A_2,...,A_n)$:
  • Probability of the intersection of n events - chain rule

Independence:

• Event A and B are independent if any of the 3 conditions hold: ^6a4bf6
  1. $P(A|B)=A$
  2. $P(B|A)=B$
  3. $P(A\cap B)=P(A).P(B)$: product rule for independent event
     • Use this to find independence: $P(A\cap B)=P(A)+P(B)-P(A\cap B)$
• Events A, B and C are independent if all following equations hold: ^a3a924
  1. $P(A,B)=P(A)P(B)$
  2. $P(B,C)=P(B)P(C)$
  3. $P(C,A)=P(C)P(A)$
  4. $P(A,B,C)=P(A)P(B)P(C)$

Conditional independence: $P(A\cap B)=P(A|B)P(B)$

• Event A and B are conditional independent given an event C:
  • $P(A,B|C)=P(A|C)P(B|C)$
• Bayes theorem