Chapter 1 Flashcards
Why is probability important for AI?
Strong mathematical foundations, including probability, are crucial for AI development.
What is probability theory?
A mathematical framework for analyzing randomness and uncertainty.
What is a random experiment?
An experiment or process with an uncertain outcome.
What are some examples of random experiments?
Throwing a die, diagnosing an illness, predicting the next U.S. president.
What is the role of probability in randomness?
Probability quantifies uncertainty and helps analyze random phenomena.
What is an interpretation of probability?
Probability can be seen as relative frequency or subjective belief.
What is relative frequency in probability?
The proportion of times an event occurs in repeated experiments.
What is subjective probability?
A personal degree of belief about the likelihood of an event.
What is a set in set theory?
A collection of elements, often represented with curly brackets.
What is a subset?
A set A is a subset of B if all elements of A are also in B.
What is the universal set in probability?
The sample space, which includes all possible outcomes.
What is the null set?
A set with no elements, denoted as ∅.
What is a Venn diagram used for?
Visualizing relationships between sets.
What is the union of two sets?
The set of elements that belong to either A or B or both, denoted A ∪ B.
What is the intersection of two sets?
The set of elements common to both A and B, denoted A ∩ B.
What is the complement of a set?
All elements in the universal set that are not in A, denoted A^c.
What are mutually exclusive sets?
Sets that do not share any elements (A ∩ B = ∅).
What is the Cartesian product of two sets?
The set of all ordered pairs (x, y) where x ∈ A and y ∈ B.
What is the multiplication principle?
If set A has M elements and set B has N elements, then A × B has M × N elements.
What is the cardinality of a set?
The number of elements in a set.
What is the inclusion-exclusion principle?
A formula to count elements in unions of overlapping sets.
What is a probability measure?
A function that assigns a probability between 0 and 1 to each event.
What are the three axioms of probability?
- P(A) ≥ 0 for all events A. 2. P(S) = 1. 3. P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
What is conditional probability?
The probability of A given that B has occurred, denoted P(A|B) = P(A ∩ B) / P(B).
What is independence in probability?
Two events A and B are independent if P(A ∩ B) = P(A)P(B).
What is the Law of Total Probability?
P(A) = Σ P(A | B_i) P(B_i), summing over a partition of the sample space.
What is Bayes’ Rule?
P(B|A) = P(A|B) P(B) / P(A), used to update beliefs based on new information.
What is an example of Bayes’ Rule application?
Medical diagnosis, where test results update the probability of a disease.
