Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
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.
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