Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for AP statistics binomial and normal? Then this AP Statistics lesson on Binomial and Normal Probability Models explains each idea step by step, with worked examples.
Quick answer: A binomial random variable counts successes in a fixed number of independent trials with the same success probability, with mean np and standard deviation √(np(1 − p)). Normal distributions model continuous bell-shaped data and, when np and n(1 − p) are at least 10, approximate binomial counts.
Standards: AP Statistics Unit 2 (2026-27) · Level: Grade 12 AP Statistics
What makes a setting binomial?
A count X of successes is binomial when each trial has two outcomes, trials are independent, the number of trials n is fixed, and the probability of success p is the same on every trial. Write X ~ B(n, p).
Sampling without replacement is treated as independent when the sample is less than 10% of the population.
How do you find binomial probabilities, the mean and the standard deviation?
P(X = k) = C(n, k) pk(1 − p)n − k
μX = np σX = √(np(1 − p))
Each seed germinates with probability 0.35, and 9 seeds are planted. Then P(X = 3) = C(9, 3)(0.35)3(0.65)6 = 0.2716, and P(X ≤ 2) = 0.3373. The mean is 9(0.35) = 3.15 and σ = √(9 × 0.35 × 0.65) ≈ 1.431.
How do you work with normal models?
A normal distribution is symmetric and bell-shaped, determined by μ and σ. Standardize with z = (x − μ)/σ, then find the area with a table or calculator (normalcdf), or use invNorm to go from an area back to a value.
Daily steps are normal with mean 8,400 and standard deviation 1,900. For 7,000 and 11,000: z = −0.74 and 1.37, so the area between them is 0.9144 − 0.2306 = 0.6838.
When can a normal curve approximate a binomial count?
- Check np ≥ 10 and n(1 − p) ≥ 10.
- Use μ = np and σ = √(np(1 − p)).
- Standardize and find the area for the event.
For n = 200 and p = 0.15, np = 30 and n(1 − p) = 170 pass the check. Then P(X ≥ 40) ≈ P(Z ≥ 1.98) = 0.0238, close to the exact binomial 0.0335.
A binomial count has a fixed number of trials. If you keep going until the first success, the number of trials is not fixed and the setting is not binomial.
- Binomial setting: BINS (binary, independent, number fixed, same p).
- P(X = k) = C(n, k)·p^k·(1 − p)^(n − k); mean np; SD √(np(1 − p)).
- Normal probabilities come from z = (x − μ)/σ and the area under the curve.
- Use invNorm to find a value from a percentile.
- Normal approximation to a binomial needs np ≥ 10 and n(1 − p) ≥ 10.
Binomial and Normal Probability Models: frequently asked questions
How do I check whether a situation is binomial?
Use BINS: two outcomes per trial, independent trials, a fixed number of trials, and the same success probability each time. If any condition fails, such as sampling without replacement from a small population, do not use the binomial formula.
What is the difference between binompdf and binomcdf?
binompdf(n, p, k) gives P(X = k), a single probability. binomcdf(n, p, k) gives P(X ≤ k), the cumulative probability. For P(X ≥ k), compute 1 − binomcdf(n, p, k − 1).
When is the normal approximation to a binomial allowed?
When both np and n(1 − p) are at least 10. Then the binomial distribution is roughly symmetric and bell-shaped, and a normal model with mean np and standard deviation √(np(1 − p)) gives a good estimate.
How do I find a value from a percentile in a normal model?
Use invNorm(area, μ, σ), or find the z-score for the area and compute x = μ + zσ. For the top 10%, the area to the left is 0.90.
Does a normal distribution have to be exactly symmetric to use the empirical rule?
The model itself is symmetric. Real data are only approximately normal, so check the histogram or a normal probability plot first. Skewed data should not be modeled by a normal distribution.