Key Idea: The binomial distribution models a situation where you repeat the same experiment n times independently, each time with the same probability p of success. It counts how many successes you get in total. The key is checking the four conditions: fixed n, constant p, independent trials, binary outcome (success/fail).
โ Binomial conditions and notation
Example: X ~ B(8, 0.4) P(X = 3) = binompdf(8, 0.4, 3) = 0.279 (3 s.f.) P(X โค 3) = binomcdf(8, 0.4, 3) = 0.594 P(X โฅ 4) = 1 โ binomcdf(8, 0.4, 3) = 1 โ 0.594 = 0.406 P(2 โค X โค 5) = binomcdf(8, 0.4, 5) โ binomcdf(8, 0.4, 1) = 0.950 โ 0.106 = 0.844 E(X) = 8 ร 0.4 = 3.2
The most common error: using binomcdf(n, p, k) for P(X โฅ k) instead of 1 โ binomcdf(n, p, kโ1). The 'โ1' is critical when working with 'at least k'. Before using binomial: check all four conditions in your working. If trials are not independent or p is not constant, the binomial model does not apply.
Paper 2 (GDC allowed): Always write the distribution statement X ~ B(n, p) first, then write the probability statement (e.g., P(X โฅ 4)), then calculate. This structure earns communication marks. Paper 1: You may need to use the formula P(X = k) = C(n,k) ร pแต ร (1โp)^(nโk) for small n. Know how to calculate C(n,k) = n!/(k!(nโk)!).