Probability Concepts
17 concepts · Grades 3-5, 6-8, 9-12 · 22 prerequisite connections
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Connected Families
Probability concepts have 20 connections to other families.
All Probability Concepts
Probability
Probability is a number between 0 and 1 (inclusive) that measures how likely an event is to occur, where 0 means impossible and 1 means certain.
Sample Space
The sample space $S$ is the set of all possible outcomes of a random experiment — every outcome that could conceivably occur.
Independent Events
Two events are independent if the occurrence of one does not change the probability of the other: $P(A \cap B) = P(A) \cdot P(B)$.
Conditional Probability
The conditional probability $P(A|B)$ is the probability of event $A$ occurring given that event $B$ has already occurred.
Randomness
Randomness is the quality of having no predictable pattern at the individual level, yet following precise probability rules over many repetitions — outcomes are uncertain one at a time but statistically regular in the long run.
Chance
Chance describes the inherent randomness in outcomes of experiments — the fact that even with complete knowledge, some events cannot be predicted with certainty.
Probability as Expectation
Probability can be interpreted as the long-run relative frequency of an event over infinitely many identical trials of a random experiment.
Events (Formal)
A formal event is a subset of the sample space — a collection of outcomes to which a probability is assigned; events can be simple (one outcome) or compound (many outcomes).
Risk
Risk is the possibility of loss or negative outcome, quantified by combining the probability of the event with the severity of its impact: Expected Loss = P(loss) times amount of loss.
Uncertainty
Uncertainty is the state of having incomplete or imperfect information about a quantity, outcome, or process, making precise prediction impossible.
Decision Under Uncertainty
Decision under uncertainty involves choosing between options whose outcomes are not known for certain, typically by comparing expected values or risk profiles.
Binomial Coefficient
The binomial coefficient $\binom{n}{k}$ counts the number of ways to choose $k$ items from $n$ distinct items without regard to order. It equals $\frac{n!}{k!(n-k)!}$.
Binomial Distribution
The probability distribution of the number of successes in $n$ independent yes/no trials, each with probability $p$.
Geometric Distribution
The probability distribution for the number of independent Bernoulli trials needed to get the first success, where each trial has success probability $p$.
Compound Probability
The probability of two or more events occurring together ($P(A \text{ and } B)$) or at least one occurring ($P(A \text{ or } B)$), accounting for whether the events are independent or dependent.
Experimental vs. Theoretical Probability
Theoretical probability is calculated from known outcomes ($P = \frac{\text{favorable}}{\text{total}}$), while experimental probability is estimated from actual trials ($P \approx \frac{\text{times event occurred}}{\text{total trials}}$). As the number of trials increases, experimental probability tends to approach theoretical probability.
Bayes' Theorem
Bayes' theorem gives the posterior probability of a hypothesis given evidence: $P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)}$.