A binomial random variable counts the successes in a fixed number of independent trials that each have the same chance of success. Check the conditions, set n and p, choose an event, and see the shaded bars, the formula, and the calculator command that gives the same answer.
Pick a situation. Check B-I-N-S first: if any condition fails, the binomial model does not apply.
Each trial ends in one of two outcomes, called "success" and "failure." Success just means the outcome you are counting, even if it is something bad like a defect.
Knowing the result of one trial does not change the probabilities for another. When sampling without replacement, use the 10% condition: n ≤ 10% of the population lets you treat trials as roughly independent.
You decide n in advance, like 10 free throws or 20 customers. If you keep going until a success, the count is not binomial.
The probability of success p is the same on every trial. A shooter who gets tired, or a deck that changes as cards are dealt, breaks this condition.
The mean is the expected number of successes. The spread is largest when p = 0.5 and shrinks as p moves toward 0 or 1.
Small p gives a right-skewed distribution and large p a left-skewed one. When np ≥ 10 and n(1 − p) ≥ 10, the shape is approximately normal.
binompdf gives one exact value; binomcdf gives "at most." Everything else is built from those two.
| Event | Command | Example: n = 10, p = 0.3 | Answer |
|---|---|---|---|
| Exactly k: P(X = 3) | binompdf(n, p, k) | binompdf(10, 0.3, 3) | 0.2668 |
| At most k: P(X ≤ 3) | binomcdf(n, p, k) | binomcdf(10, 0.3, 3) | 0.6496 |
| Fewer than k: P(X < 3) | binomcdf(n, p, k − 1) | binomcdf(10, 0.3, 2) | 0.3828 |
| At least k: P(X ≥ 3) | 1 − binomcdf(n, p, k − 1) | 1 − binomcdf(10, 0.3, 2) | 0.6172 |
| More than k: P(X > 3) | 1 − binomcdf(n, p, k) | 1 − binomcdf(10, 0.3, 3) | 0.3504 |
| Between: P(2 ≤ X ≤ 4) | binomcdf(n, p, 4) − binomcdf(n, p, 1) | 0.8497 − 0.1493 | 0.7004 |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 2.10. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 2 terminology guide.