Binomial Distribution

Unit 4 · Probability, Random Variables & Distributions

What AP Stats asks here

The binomial distribution counts successes in a fixed number of independent Bernoulli trials. The recurring AP gates are BINS — Binary outcomes, Independent trials, fixed N, Same probability — and the easy confusion with geometric (which leaves N variable and counts trials until the first success).

BINS conditions

Binary

\text{success or failure on each trial}

Independent

\text{trials do not influence each other}

Number fixed

n \text{ is set in advance}

Same probability

p \text{ constant on every trial}

Binomial formulas

PMF

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

Mean

\mu_X = np

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

AP Tip: Use the complement for 'at least one'. P(at least one) = 1 − P(none) = 1 −

(1 - p)^n

is one step; summing all the positive cases is many steps.

Caution: Without replacement violates independence. The 10% rule of thumb lets you treat the count as approximately binomial when the sample is less than 10% of the population.

Type 1

Verify BINS conditions

Run each condition explicitly: is it binary, independent, fixed N, same p? Any violation rules out binomial.

Example 1

Roll a die until you get a 6; let X = number of rolls. Is X binomial? (A) Yes — the trials are independent. (B) No — N is not fixed; this is geometric. (C) Yes — it's binomial with p = 1/6. (D) No — the outcomes are not binary.

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Type 2

Compute $P(X = k)$ or $P(X \le k)$

Plug into $\binom{n}{k} p^k (1-p)^{n-k}$ or use binomcdf. Watch the boundary: $P(X \le k)$ includes $k$ .

Example 2

A factory produces 5% defective items. In a sample of 20, what is

P(\text{at least one defective})

? (A) 0.05 (B) 0.36 (C) 0.64 (D) 1.00

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Type 3

Mean and SD

$\mu = np$ and $\sigma = \sqrt{np(1-p)}$ . These are the routine binomial summaries.

Example 3

For

X \sim \text{Bin}(200, 0.4)

, what are

\mu

and

\sigma

? (A)

\mu = 80, \sigma \approx 6.93

(B)

\mu = 40, \sigma \approx 6.93

(C)

\mu = 80, \sigma = 48

(D)

\mu = 80, \sigma \approx 8.94

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Binomial Distribution

What AP Stats asks here

BINS conditions

Binomial formulas

Verify BINS conditions

Compute P(X=k)P(X = k)P(X=k) or P(X≤k)P(X \le k)P(X≤k)

Mean and SD

Compute $P(X = k)$ or $P(X \le k)$