Lecture 19 Fundamental of Probability Theory
- Basic Definition and Rules
- Probability function assigns a real number to each outcome
- Range of probability
- Equiprobability rule \(P(A_{i}) = \dfrac{1}{n}\)
- Mutually exclusive: only one of them can happen at a time
- "Or": \(P(A \cup B) = P(A) + P(B)\)
- "Everything": \(\sum P(A_{i}) = 1\)
- "Not": \(P(\neg A) = 1- P(A)\)
- Independent: occurrence of one doesn't affect the occurrence of another
- "And": \(P(A\cap B) = P(A) \times P(B)\)
- Conditional Probability
- Given that some other thing happens
- \(P(A | C) = \dfrac{P(A\cap B)}{P(B)}\)
- The General AND Rule
- \(P(A|B) \times P(B) = P(B|A) \times P(A)\)
- Independence Conditions
- \(P(A|B) = P(A)\)
- \(P(B|A) = P(B)\)
- \(P(A\cap B) = P(A) \times P(B)\)
- Total Probability Rule
- \(P(D) = P(D \cap B) \cup [D \cap \neg B])\)
- \(P(D)=P(D \cap B) + P(D \cap \neg B)\)
- and \(P(D \cap B) = P(D|B) \times P(B)\)
- \(\implies\) \(P(D) = P(D|B) \times P(B) + P(D|\neg B) \times P(\neg B)\)
- Bayes' Rule or Bayes' Theorem ^96fb64
- \(P(B|D) = \dfrac{P(D|B)\times P(B)}{P(D)}\)
- And use the total probability rule to calculate: \(P(D)\)