Discrete Probability Models in AP Statistics: Binomial and Geometric

Binomial distribution

A discrete probability model for the number of successes in a fixed number of identical trials when the binomial conditions are met.

Binomial setting (BINS)

Conditions for a binomial model: Binary outcomes, Independent trials, Number of trials n fixed, Same probability of success p each trial.

Binary condition

Each trial has exactly two outcomes, typically labeled success and failure.

Independence (binomial/geometric)

The outcome of one trial does not affect the outcome of another trial.

Number condition (binomial)

The number of trials n is fixed in advance (you do not keep going “until” something happens).

Same probability condition

The probability of success p stays constant from trial to trial.

10% condition

When sampling without replacement, trials can be treated as approximately independent if the sample size is no more than 10% of the population.

Binomial random variable (X)

The count of successes in n trials; possible values are 0, 1, 2, …, n.

Binomial notation

X ~ Bin(n, p), where n is the number of trials and p is the probability of success on each trial.

Binomial probability mass function (pmf)

P(X = k) = (n choose k) p^k (1 − p)^(n − k), for k = 0, 1, …, n.

Combination term (n choose k)

The number of ways to arrange k successes among n trials; accounts for “any order” of successes and failures.

Common binomial mistake: missing combinations

Using only p^k(1−p)^(n−k) ignores the number of sequences and is only correct for one specific sequence (order fixed).

Binomial mean

For X ~ Bin(n,p), the mean (expected number of successes) is μ = np.

Binomial standard deviation

For X ~ Bin(n,p), the standard deviation is σ = √(np(1−p)).

Complement rule (binomial)

A strategy for “at least/at most” probabilities, e.g., P(X ≥ 1) = 1 − P(X = 0).

Inequality language (AP Stats)

Phrases translate to symbols: “at least” means ≥, “at most/no more than” means ≤, and “between” is usually inclusive unless stated otherwise.

Binomial pdf (calculator)

Technology function that computes an exact probability P(X = k) for a binomial random variable.

Binomial cdf (calculator)

Technology function that computes a cumulative probability P(X ≤ k) for a binomial random variable.

Geometric distribution

A discrete probability model for the number of trials needed to get the first success in repeated independent trials with constant success probability p.

Geometric setting (conditions)

Repeated independent trials, each trial is success/failure, p stays the same, and X counts trials until the first success.

Geometric notation

X ~ Geom(p), typically with X defined in AP Statistics as the number of trials until the first success (values 1, 2, 3, …).

Geometric pmf

If X counts trials until first success, then P(X = k) = (1 − p)^(k−1) p.

Off-by-one error (geometric)

A common mistake is writing (1−p)^k p instead of (1−p)^(k−1) p, forgetting there are only k−1 failures before the first success on trial k.

Geometric “greater than” probability

P(X > k) = (1 − p)^k, since it means no successes in the first k trials.

Memoryless property (geometric)

Past failures don’t change future waiting probabilities: P(X > m+n | X > m) = P(X > n), assuming independence and constant p.