Statistics · Unit 4: Probability, Random Variables, and Probability Distributions · 14 min read · Updated 2026-05-11
Introduction to the Binomial Distribution — AP Statistics
AP Statistics · Unit 4: Probability, Random Variables, and Probability Distributions · 14 min read
1. Binomial Setting and Binomial Random Variables★★☆☆☆⏱ 4 min
The binomial distribution is the most commonly tested discrete probability distribution on the AP Statistics exam, describing the number of successes in a fixed set of independent binary trials, each with constant probability of success.
**B**: Binary outcomes: each trial has exactly two outcomes (success = outcome we count, failure = the other outcome)
**I**: Independent trials: the outcome of one trial does not affect any other trial
**N**: Fixed Number of trials: $n$ is set before data collection, does not depend on trial outcomes
**S**: Same probability of success: $p$ is constant across all trials
Exam tip: On the AP exam, you must explicitly name and verify all four BINS conditions to earn full credit on verification questions — skipping one condition will cost you a point.
2. Calculating Binomial Probabilities★★★☆☆⏱ 4 min
Once you confirm a setting is binomial, you can calculate the probability of getting exactly $k$ successes using the binomial probability formula, which combines combinations (counting the number of possible success sequences) and the probability of each sequence.
The combination formula counts the number of ways to choose positions for $k$ successes out of $n$ total trials:
\binom{n}{k} = \frac{n!}{k! (n-k)!}
Each sequence of $k$ successes and $(n-k)$ failures has a probability of $p^k (1-p)^{n-k}$ for independent trials. Combining these gives the binomial probability mass function:
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
For cumulative probabilities (ranges of outcomes), sum individual probabilities: $P(X \leq k) = \sum_{i=0}^k P(X=i)$. For probabilities of at least $k$ successes, use the complement rule: $P(X \geq k) = 1 - P(X \leq k-1)$.
Exam tip: You can use your AP-approved calculator to get probabilities directly, but you must still write down the binomial probability formula with substituted values of $n$, $k$, and $p$ to earn full credit on FRQs.
3. Mean, Variance, and Standard Deviation★★☆☆☆⏱ 3 min
A binomial random variable is the sum of $n$ independent Bernoulli trials, each with mean $p$ and variance $p(1-p)$. This gives simple closed-form formulas for the mean, variance, and standard deviation, so you do not need to construct a full probability table.
The mean (expected number of successes) is:
\mu_X = E(X) = np
Intuitively, if you have $n$ trials each with probability $p$ of success, you expect an average of $np$ successes. The variance and standard deviation are:
\sigma_X^2 = Var(X) = np(1-p)
\sigma_X = SD(X) = \sqrt{np(1-p)}
Exam tip: Always interpret the expected value in context if asked: for this example, *"If we observed many days with 115 customers, we would expect an average of 32.2 customers to order a dairy alternative"*.
4. AP-Style Practice Problems★★★☆☆⏱ 4 min
Common Pitfalls
Why: Students confuse stopping rules for geometric distributions with the fixed n requirement for binomial distributions.
Why: Students assume independence always holds, but sampling without replacement changes p between trials.
Why: Students forget that "at least 2" excludes all values less than 2, not all values less than or equal to 2.
Why: Students misremember which number corresponds to which value.
Why: Students mix up formulas and do not read the question carefully.