## Probability

## AI, Decision Making, and Probability

About decision-making

Uncertainty, likelihood, and probability

Decision-theoretic methods

## Probability

Idea:

*chance*and*likelihood*are important concepts for real reasoningMethod: assign probabilities to events and combinations of events

Reason from model using calculation

This is a general plan:

`Evidence -----------------> Conclusions | ^ | | v | Model -----------------> Extrapolation`

## Probability of Events

Domain: events

*pr(E)*is probability that event*E*happens*pr(E) = #E / (#E + #¬E)*For a coin

*pr(H) = #H / (#H + #T) = 1 / 2*For a pair of dice:

*pr(R7) = #R7 / #R**#R7 = #{(1, 6), (2, 5), (3, 4), …} = 6**#R = #{D×D} = 36**pr(R7) = 6 / 36 = 1/6*Check with computer program prob7

## Probability of Logical Situations

Domain: propositional formula

*pr(p)*is probability of logical combination of events*p*is a prop formula

Priors and conditionals

*pr(p|q)*is prob of*p*given*q*(easy to get backward)

## Axioms Of Probability

equivalence: if

*p ≡ q*then*pr(p) = pr(q)*range:

*0 ≤ pr(p) ≤ 1*negation:

*pr(¬p) = 1 - pr(p)*conjunction:

*pr(p ∧ q) = pr(p) pr(q|p) = pr(q) pr(p|q)*

## Derived Probability Rules

disjunction:

*pr(p ∨ q) = pr(¬(¬p ∧ ¬q)) = 1 - pr(¬p ∧ ¬q)*Bayes's Rule:

`pr(q) pr(p|q) = pr(p ∧ q) = pr(p) pr(q|p) pr(q) pr(p|q) = pr(q|p) pr(p) pr(p|q) = pr(q|p) pr(p) / pr(q)`

## Independence

When

*p*and*q*are conditionally independent*pr(p|q) = pr(p)*By Bayes's Rule

`pr(q|p) = pr(p|q) pr(q) / pr(p) = pr(p) pr(q) / pr(p) = pr(q)`

In this case, conjunction gets easier

*pr(p ∧ q) = pr(p|q) pr(q) = pr(p) pr(q)*

When

*p*and*q*are strictly independent*pr(p ∧ q) = 0*In this case, disjunction gets easier

`pr(p ∨ q) = 1 - pr(¬p ∧ ¬q) = 1 - pr(¬p) pr(¬q) = 1 - (1 - pr(p))(1 - pr(q)) = 1 - (1 - pr(p) - pr(q) - pr(p) pr(q)) = pr(p) + pr(q) + pr(p) pr(q) = pr(p) + pr(q)`