6.3 Probability, Counting, and Distributions

Probability Starts With the Event

The AEPA Domain V probability competency includes simple, compound, and conditional probability; counting principles; graphical probability models; simulations; and uniform, binomial, and normal distributions. That is a broad list, but the core habit is consistent: define the sample space and the event before calculating. Many wrong answers come from using a correct-looking formula on the wrong event.

A sample space is the set of all possible outcomes. An event is a subset of that space. The probability of an event is favorable outcomes divided by total outcomes when outcomes are equally likely. The complement of event A is not A, so P(not A) = 1 - P(A). The union A or B includes outcomes in A, in B, or in both. The intersection A and B includes outcomes shared by both events.

Core Probability Rules

Mutually exclusive and independent are not the same. Two nonempty mutually exclusive events cannot both occur, so knowing one occurred makes the other impossible. Independent events do not affect each other's probabilities and can occur together. This distinction is a common certification-level misconception because both phrases sound like "unrelated," but mathematically they describe different structures.

Counting Before Probability

Use the fundamental counting principle when a process has stages: multiply the number of choices at each stage. Use a permutation when order matters, such as arranging officers president, secretary, and treasurer. Use a combination when order does not matter, such as choosing a committee of three. The quick diagnostic is whether rearranging the same selected people creates a new outcome. If yes, order matters. If no, use combinations.

Worked example: From 8 students, how many 3-person committees can be chosen? The selection {A, B, C} is the same committee as {C, B, A}, so order does not matter. The count is 8C3 = 8! / (3!5!) = 56. If the question instead asks for president, secretary, and treasurer, order matters because A as president differs from A as secretary. The count is 8P3 = 8 x 7 x 6 = 336.

Conditional Probability and Tables

Two-way tables are frequent probability models. Suppose 60 students are surveyed: 35 take statistics, 25 do not; 18 of the statistics students also take computer science, and 7 non-statistics students take computer science. If a student is chosen at random given that the student takes computer science, the denominator is not 60. The condition restricts the sample space to the 25 computer science students. The probability that the student also takes statistics is 18 / 25. A distractor such as 18 / 60 answers a joint probability, not a conditional probability.

Distribution Recognition

A uniform distribution assigns equal probability across outcomes or intervals. A fair die is discrete uniform because each face has probability 1/6. A binomial distribution applies when there are fixed independent trials, each trial has two outcomes, the probability of success is constant, and the random variable counts successes. The probability of exactly k successes in n trials is nCk p^k (1 - p)^(n-k). A normal distribution is continuous, symmetric, and bell-shaped. Its mean and standard deviation determine center and spread, and symmetry is often enough for AEPA-level reasoning.

A simulation must match the event. To model free throws with a 70% success rate, a random digit simulation could assign digits 0 through 6 as makes and 7 through 9 as misses, then group digits by the number of attempts. If the real question asks for the probability of the first make occurring on the third attempt, the simulation must stop after sequences miss, miss, make. If it asks for exactly three makes in five attempts, the simulation must use five-trial groups. Same success probability, different event.

Common Traps

At least one success is usually easier through the complement: 1 minus the probability of no successes. Without replacement usually creates dependence because the sample space changes after each draw. With replacement usually preserves independence. Normal curve questions often use symmetry: if 12% lie below a point equally far below the mean as another point is above the mean, then 12% lie above the upper point. Finally, do not round too early; small probabilities in binomial settings can shift answer choices.

Probability also connects to expected value, even when the official wording emphasizes distributions. Expected value is the long-run average outcome: multiply each numerical outcome by its probability and add the products. It is not necessarily an outcome that can occur in one trial. A raffle may have an expected value of negative two dollars for the player even though no ticket literally pays negative two dollars; the value summarizes the average loss over many plays.