Fundamentals of Probability

Fundamentals of Probability

• Probability theory defines uncertainty related to outcomes, such as coin flips or weather predictions.
• A random variable represents possible outcomes with assigned probabilities; for a fair coin, P(heads) = 50%.
• Probability ranges from 0 (impossible outcome) to 1 (certain outcome); values outside this range are invalid.

Conditional and Unconditional Probabilities

• Conditional probability, P(A | B), is the probability of event A occurring given that event B has occurred.
• Unconditional (marginal) probability refers to the likelihood of an event occurring without any conditions applied.
• Joint probability, P(AB), represents the probability of two events A and B occurring simultaneously.

Events and Event Spaces

• An event is defined as a specific outcome or a combination of outcomes (e.g., rolling a 3 on a die).
• Event space encompasses all potential outcomes for a random variable, such as {1, 2, 3, 4, 5, 6} for a six-sided die.
• Total probability for all outcomes in an event space equals 1.

Characteristics of Events

• Independent events yield probabilities that do not influence one another; knowledge of one event does not change the other's probability.
• If A and B are independent, then P(A) × P(B) = P(AB) and P(A | B) = P(A).
• Example of independent events: flipping a coin multiple times; the outcome of the first flip does not affect the second.

Mutually Exclusive Events

• Mutually exclusive events cannot occur simultaneously; if one happens, the other cannot.
• Example: rolling a die and getting an even number versus rolling a three—these outcomes cannot happen at the same time.
• For mutually exclusive events A and B, P(A ∪ B) = P(A) + P(B); outcomes are additive.