Algebra 2 · Unit 11 · Topic 11.5
Binomial Trials and Expected Counts
A binomial model counts successes in n independent trials with two outcomes and the same success probability p on each trial.
Definition and explanation
A binomial model counts successes in n independent trials with two outcomes and the same success probability p on each trial. It gives P(X = k) = C(n, k)pᵏ(1 − p)^(n − k), and expected count E(X) = np.
Worked example
For 4 independent fair coin tosses, the probability of exactly 3 heads is C(4, 3)(1/2)³(1/2) = 4/16 = 1/4. The expected number of heads is 4(1/2) = 2, though an individual set need not have exactly 2.
Common mistake
Changing success probabilities or dependent trials can invalidate the binomial model.
Key takeaway
A binomial model counts successes in n independent trials with two outcomes and the same success probability p on each trial.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.