Probability

Probability assesses the likelihood of an event in a situation that involves randomness. It quantifies the degree of certainty that an event will happen as a number from 0 through 1. This number is generally interpreted as the relative frequency of occurrence of the event over the long run. The structure of a probability model begins by listing or describing the possible outcomes for a random situation (the sample space) and assigning probabilities based on an assumption about long-run relative frequency. In situations such as flipping a coin, rolling a number cube, or drawing a card, it is reasonable to assume various outcomes are equally likely. Compound events constructed from these simple ones can be represented by tree diagrams and by frequency or relative frequency tables. The probabilities of compound events can be computed using these representations and by applying the additive and multiplicative laws of probability. Interpreting these probabilities relies on an understanding of independence and conditional probability, approachable through the analysis of two-way tables. Converting a verbally-stated problem into the symbols and relations of probability requires careful attention to words such as //and//, //or//, //if//, and //all//, and to grammatical constructions that reflect logical connections. This is especially true when applying probability models to real-world problems, where simplifying assumptions are also usually necessary in order to gain at least an approximate solution. //Connections to Statistics and Expressions//. Probability is the foundation for drawing valid conclusions from sampling or experimental data. Counting has an advanced connection with Expressions through Pascal's triangle and binomial expansions.

Students understand that:
 * Core Concepts**
 * 1) Probability models outcomes for situations in which there is inherent randomness, quantifying the degree of uncertainty in terms of relative frequency of occurrence.
 * 2) The law of large numbers provides the basis for estimating certain probabilities by use of empirical relative frequencies.
 * 3) The laws of probability govern the calculation of probabilities of combined events.
 * 4) Interpreting probabilities contextually is essential to rational decision-making in situations involving randomness.

Students can and do:
 * Core Skills**
 * 1) Compute theoretical probabilities by systematically counting points in the sample space.
 * 2) Interpret probabilities of compound events using concepts of independence and conditional probability.
 * 3) Compute probabilities of compound events.
 * 4) Estimate probabilities empirically.
 * 5) Identify and explain common misconceptions regarding probability.
 * 6) Adapt probability models to solve real-world problems.