Use a binomial distribution when you count successes in a fixed number of trials. Use a Poisson distribution when you count events at an average rate over an interval, with no fixed number of trials. Use a normal distribution for a measured quantity such as mass or time. Long questions often feed the answer from one into another.
B(n, p)Po(λ), with λ scaled to the intervalN(μ, σ²)A typical long question has two layers:
B(n, p), using the p from the inner layer.A third part may then add a conditional probability. Write down, for each part, what is being counted and out of how many; that alone usually tells you the distribution.
The number of typing errors on a page follows a Poisson distribution with mean 0.5. (a) Find the probability that a page has no errors. (b) Find the probability that exactly 8 pages out of 10 have no errors.
X ~ Po(0.5): P(X = 0) = e−0.5 = 0.6065Y ~ B(10, e−0.5)P(Y = 8) = C(10, 8)(e−0.5)⁸(1 − e−0.5)²= 45 × e−4 × (0.39347)² = 0.1276Answer: (a) 0.6065; (b) 0.1276
Each item produced is defective with probability 0.05, independently. Find the probability that a box of 20 items contains at most 1 defective item.
X ~ B(20, 0.05)P(X ≤ 1) = (0.95)²⁰ + 20(0.05)(0.95)¹⁹= 0.3585 + 0.3774 = 0.7358Answer: 0.7358
e−0.5, not 0.61) in the outer binomial. Rounding early can change the fourth decimal place of the final answer.How to study: Click any question below to instantly load it on the coordinate grid and check your steps.