Concept Lesson

Binomial Distribution: Conditions, Formula, Mean, and Calculator Steps

A lesson in the binomial distribution that moves from intuition and definitions to worked reasoning, error correction, and independent practice.

Course status: Revised 2026-27 course
Updated: July 18, 2026
Practice: Easy, Tough and Toughest

Lesson Goals: Binomial Distribution

A binomial variable counts successes in a fixed number of binary, independent trials with constant success probability, and its mean and standard deviation follow from n and p.

Reader taskBINS conditions, exact and cumulative probabilities, mean, standard deviation, and calculator use

Planned modules7

Mathematics3 expressions

Worked checks60

Boundary: Geometric material is legacy and belongs in P46-P48.

BINS conditions

BINS conditions in binomial distribution: The exact probability is 0.09783; the model mean is 4.290. Across many repetitions of 11 trials, the average number of successes approaches $np=4.290$ .

Worked reasoning

For BINS conditions in binomial distribution, In a constructed BINS setting for a water-filtration experiment, $n=11$ , $p=0.39$ , and X counts successes. Find $P(X=2)$ for bins conditions.

P(X=2)=\binom{11}{2}(0.39)^2(0.61)^{9}=0.09783.

When the idea is valid

For BINS conditions in binomial distribution, Verify binary outcomes, independent trials, fixed n, and constant success probability before using the binomial model.

Misconception to remove

For BINS conditions in binomial distribution, reject this error: Do not omit the combination factor; it counts the different orders producing exactly k successes.

Binomial probability formula

Binomial probability formula in binomial distribution: The exact probability is 0.23235; the model mean is 3.720. Across many repetitions of 12 trials, the average number of successes approaches $np=3.720$ .

Worked reasoning

For Binomial probability formula in binomial distribution, In a constructed BINS setting for a quality-control inspection, $n=12$ , $p=0.31$ , and X counts successes. Find $P(X=3)$ for binomial probability formula.

P(X=3)=\binom{12}{3}(0.31)^3(0.69)^{9}=0.23235.

When the idea is valid

For Binomial probability formula in binomial distribution, Verify binary outcomes, independent trials, fixed n, and constant success probability before using the binomial model.

Misconception to remove

For Binomial probability formula in binomial distribution, reject this error: Do not omit the combination factor; it counts the different orders producing exactly k successes.

Exact, at-most, and at-least probabilities

Exact, at-most, and at-least probabilities in binomial distribution: The exact probability is 0.05718; the model mean is 7.200. Across many repetitions of 16 trials, the average number of successes approaches $np=7.200$ .

Worked reasoning

For Exact, at-most, and at-least probabilities in binomial distribution, In a constructed BINS setting for a classroom memory study, $n=16$ , $p=0.45$ , and X counts successes. Find $P(X=4)$ for exact, at-most, and at-least probabilities.

P(X=4)=\binom{16}{4}(0.45)^4(0.55)^{12}=0.05718.

When the idea is valid

For Exact, at-most, and at-least probabilities in binomial distribution, Verify binary outcomes, independent trials, fixed n, and constant success probability before using the binomial model.

Misconception to remove

For Exact, at-most, and at-least probabilities in binomial distribution, reject this error: Do not omit the combination factor; it counts the different orders producing exactly k successes.

Mean and standard deviation

Mean and standard deviation in binomial distribution: The exact probability is 0.20632; the model mean is 5.270. Across many repetitions of 17 trials, the average number of successes approaches $np=5.270$ .

Worked reasoning

For Mean and standard deviation in binomial distribution, In a constructed BINS setting for a package-delivery sample, $n=17$ , $p=0.31$ , and X counts successes. Find $P(X=5)$ for mean and standard deviation.

P(X=5)=\binom{17}{5}(0.31)^5(0.69)^{12}=0.20632.

When the idea is valid

For Mean and standard deviation in binomial distribution, Verify binary outcomes, independent trials, fixed n, and constant success probability before using the binomial model.

Misconception to remove

For Mean and standard deviation in binomial distribution, reject this error: Do not omit the combination factor; it counts the different orders producing exactly k successes.

Calculator commands

Calculator commands in binomial distribution: The exact probability is 0.03645; the model mean is 5.610. Across many repetitions of 17 trials, the average number of successes approaches $np=5.610$ .

Worked reasoning

For Calculator commands in binomial distribution, In a constructed BINS setting for a greenhouse germination experiment, $n=17$ , $p=0.33$ , and X counts successes. Find $P(X=2)$ for calculator commands.

P(X=2)=\binom{17}{2}(0.33)^2(0.67)^{15}=0.03645.

When the idea is valid

For Calculator commands in binomial distribution, Verify binary outcomes, independent trials, fixed n, and constant success probability before using the binomial model.

Misconception to remove

For Calculator commands in binomial distribution, reject this error: Do not omit the combination factor; it counts the different orders producing exactly k successes.

Worked examples

Worked examples in binomial distribution: The exact probability is 0.11441; the model mean is 5.180. Across many repetitions of 14 trials, the average number of successes approaches $np=5.180$ .

Worked reasoning

For Worked examples in binomial distribution, In a constructed BINS setting for a water-filtration experiment, $n=14$ , $p=0.37$ , and X counts successes. Find $P(X=3)$ for worked examples.

P(X=3)=\binom{14}{3}(0.37)^3(0.63)^{11}=0.11441.

When the idea is valid

For Worked examples in binomial distribution, Verify binary outcomes, independent trials, fixed n, and constant success probability before using the binomial model.

Misconception to remove

For Worked examples in binomial distribution, reject this error: Do not omit the combination factor; it counts the different orders producing exactly k successes.

Common errors

Common errors in binomial distribution: The exact probability is 0.04752; the model mean is 7.280. Across many repetitions of 14 trials, the average number of successes approaches $np=7.280$ .

Worked reasoning

For Common errors in binomial distribution, In a constructed BINS setting for a city bus arrival investigation, $n=14$ , $p=0.52$ , and X counts successes. Find $P(X=4)$ for common errors.

P(X=4)=\binom{14}{4}(0.52)^4(0.48)^{10}=0.04752.

When the idea is valid

For Common errors in binomial distribution, Verify binary outcomes, independent trials, fixed n, and constant success probability before using the binomial model.

Misconception to remove

For Common errors in binomial distribution, reject this error: Do not omit the combination factor; it counts the different orders producing exactly k successes.

Formula and Notation Reference

Exact binomial probability

P(X=k)=\binom nk p^k(1-p)^{n-k}

Exact binomial probability in Binomial Distribution: Define the event or random variable first; complements, conditioning, and trial assumptions determine which probability expression applies.

Binomial mean

\mu_X=np

Binomial mean in Binomial Distribution: This expression belongs specifically to the binomial distribution; define every symbol and apply the scope rule for BINS conditions, exact and cumulative probabilities, mean, standard deviation, and calculator use before calculation.

Binomial standard deviation

\sigma_X=\sqrt{np(1-p)}

Binomial standard deviation in Binomial Distribution: Define success, the population proportion, the sample proportion, and the denominator before substituting values.

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Guided, Independent and Challenge Practice

Every question in Binomial Distribution: Conditions, Formula, Mean, and Calculator Steps is newly written from the revised framework and the logic visible in public College Board materials. Constructed numerical settings are identified as instructional scenarios and are never represented as measurements from a real population. No released or secure question wording is reproduced.

Easy Practice

Easy 1: Calculator commands

Question P45-Easy-1. In a constructed BINS setting for a water-filtration experiment, $n=13$ , $p=0.54$ , and X counts successes. Find $P(X=2)$ for calculator commands.