AI Decision Making under Uncertainty

Representing Uncertainty with Probability

• Partial knowledge of the world leads to uncertainty, making it essential to make optimal decisions with limited information.
• Probability uncertainty can be represented as a number of events and their likelihood.

Possible Worlds and Probability Notation

• Every possible situation can be thought of as a world, represented by the lowercase Greek letter omega (ω).
• The probability of a certain world is represented as P(ω).

Axioms in Probability

• 0 < P(ω) < 1: every value representing probability must range between 0 and 1.
• Zero represents an impossible event, while one represents a certain event.
• The higher the value, the more likely the event is to happen.
• The probabilities of every possible event, when summed together, are equal to 1.

Summing Probabilities

• The probability of rolling a number R with a standard die is P(R) = 1/6, due to six equally likely possible worlds.
• When rolling two dice, there are 36 possible events, which are equally likely to happen.

Calculating Probability

• To get the probability of an event, divide the number of worlds in which it occurs by the number of total possible worlds.
• Example: P(12) = 1/36, the probability of rolling two dice and getting a sum of 12.
• Example: P(7) = 6/36 = 1/6, the probability of rolling two dice and getting a sum of 7.

Unconditional and Conditional Probability

• Unconditional probability is the degree of belief in a proposition in the absence of any other evidence.
• Conditional probability is the degree of belief in a proposition given some evidence that has already been revealed.
• Conditional probability is expressed as P(a | b), meaning "the probability of event a occurring given that we know event b to have occurred."

Conditional Probability Formula

• P(a | b) = P(a, b) / P(b): the probability of a given b is equal to the probability of a and b being true, divided by the probability of b.