In AP Statistics, the probability of success, written p, is the chance that a single trial of a binary (success/failure) process results in a success; it stays constant across independent trials and appears in both the binomial formula P(X=x)=C(n,x)p^x(1-p)^(n-x) and the geometric formula P(X=x)=(1-p)^(x-1)p.
The probability of success, denoted p, is the chance that any single trial of a two-outcome process turns out to be a "success." Its partner is the probability of failure, 1 − p. "Success" here doesn't mean something good. It just means the outcome you're counting. If you're tracking defective electronic components and each one has a 0.12 chance of being defective, then p = 0.12 even though a defect is bad news.
In Unit 4, p is the engine inside two distributions. For a binomial random variable, p plugs into P(X = x) = C(n, x)·p^x·(1−p)^(n−x) to find the probability of exactly x successes in n independent trials. For a geometric random variable, p plugs into P(X = x) = (1−p)^(x−1)·p to find the probability that the first success lands on trial x. In both cases, the key assumption is that p is the same on every trial and the trials are independent. If a free-throw shooter makes 70% of her shots, p = 0.7 on shot one, shot two, and shot eight.
Probability of success lives in Unit 4: Probability, Random Variables, and Probability Distributions, specifically Topics 4.10 through 4.12. It's baked into nearly every learning objective there. You use p to calculate binomial probabilities (AP Stats 4.10.B), to estimate them with simulations (AP Stats 4.10.A), and to find binomial parameters like the mean μ = np and standard deviation √(np(1−p)) (AP Stats 4.11.A). On the geometric side, p gives you the probability of the first success on a given trial (AP Stats 4.12.A) and the parameters μ = 1/p and σ = √(1−p)/p (AP Stats 4.12.B). If you misidentify p in a problem, every calculation after it is wrong, which is why the exam loves stems that make you pull p out of a word problem yourself. It also sets up the bigger picture later in the course, since p is a population parameter you'll eventually estimate and test with inference.
Keep studying AP Statistics Unit 4
Binomial Distribution (Unit 4)
A binomial random variable counts successes in n independent trials, each with the same probability of success p. The two numbers n and p completely define the distribution, which is why you'll see notation like X ~ Bin(8, 0.7). Know p and n, and you know everything.
Geometric Distribution (Unit 4)
Same p, different question. Instead of counting successes in a fixed number of trials, a geometric variable asks which trial gives you the first success. A small p means a long expected wait, since the geometric mean is 1/p. If p = 0.25, you expect the first success around trial 4.
Bernoulli Trial (Unit 4)
A Bernoulli trial is one single success/failure attempt with probability of success p. Binomial and geometric settings are just stacks of Bernoulli trials. The whole framework only works if p stays constant from trial to trial.
Expected Value (Unit 4)
p directly drives the center of both distributions. A binomial mean is np (8 free throws at p = 0.7 means about 5.6 makes on average), and a geometric mean is 1/p. Changing p shifts where the distribution sits.
Multiple-choice questions usually hand you p inside a context and make you use it. A classic stem reads like "a basketball player makes 70% of her free throws; she attempts 8; find the probability she makes exactly 6." Your job is to recognize p = 0.7, n = 8, and apply the binomial formula. Other MCQs test whether you know the conditions that make p usable, like the question asking which assumption is NOT required for the binomial formula (constant p and independent trials are required; equal p across trials trips people up). Simulation questions also show up, where you estimate a probability involving p = 0.25 from 300 simulated trials instead of computing it exactly, matching AP Stats 4.10.A. On the FRQ side, the 2025 exam (Q3, the restaurant playlist problem) built on this exact setup. Expect to define the random variable, identify p in context, calculate a probability, and interpret your answer with units and context, because interpretation is its own scored skill (AP Stats 4.11.B and 4.12.C).
p is the input and P(X = x) is the output. The probability of success p describes one single trial (a 0.7 chance of making one free throw). P(X = x) is what the binomial or geometric formula spits out for a whole scenario (the chance of exactly 6 makes in 8 attempts, which is about 0.296). On the exam, p comes straight from the problem's wording, while P(X = x) is the thing you compute. Mixing them up leads to plugging the wrong number into formulas like np or 1/p.
The probability of success p is the chance that one single trial results in a success, and "success" just means the outcome you're counting, even if it's something bad like a defect.
p must stay constant across all trials, and trials must be independent, for the binomial and geometric formulas to be valid.
In the binomial setting, p appears in P(X = x) = C(n, x)p^x(1−p)^(n−x), and in the geometric setting it appears in P(X = x) = (1−p)^(x−1)p.
p controls the parameters of both distributions, with a binomial mean of np and standard deviation √(np(1−p)), and a geometric mean of 1/p and standard deviation √(1−p)/p.
On the exam, you pull p directly from the context (like "makes 70% of free throws" means p = 0.7), then calculate and interpret in context with appropriate units.
It's the chance, written p, that a single trial of a success/failure process ends in success. It's one of the two parameters of a binomial distribution (along with n) and the only parameter of a geometric distribution.
No. Success is just the outcome you're counting. If a quality control engineer tracks defective components with a 0.12 defect rate, then p = 0.12 even though a defect is the bad result.
p describes one trial and comes straight from the problem (like 70% free throw shooting means p = 0.7). P(X = x) describes a whole scenario and is what you compute with the formula, like the roughly 0.296 chance of exactly 6 makes in 8 attempts.
No. A core binomial condition is that p is identical on every one of the n independent trials. If p changes between trials, the binomial formula doesn't apply.
For a binomial variable the mean is np, so 8 free throws at p = 0.7 averages 5.6 makes. For a geometric variable the mean is 1/p, so with p = 0.25 you expect the first success around trial 4.