Applications of the binomial and the Poisson distributions: แบบฝึกหัด

It is known that \(2\%\) of the items produced by a factory are defective. If a sample of \(10\) items is randomly selected, find the probability that at most \(1\) item is defective, correct to 4 decimal places.

A machine produces components of which \(1\%\) are defective. A batch of \(400\) components is packed in a box. Using a suitable approximation, find the probability that a box contains more than \(6\) defective components, correct to 4 decimal places.

A certain medicine has a 0.5% chance of causing a specific side effect. If 800 patients are randomly selected to take the medicine, use a Poisson approximation to find the probability that at most 1 patient experiences the side effect.

The number of accidents occurring at a certain road junction follows a Poisson distribution with a mean of \(0.2\) per day. Find the probability that there is at least one day with accidents in a \(5\)-day week.

The mean and variance of a binomial distribution representing defective items are \(0.05\) and \(0.0495\) respectively. Find the probability \(p\) that an item is defective and, using a Poisson approximation, calculate the probability that there are more than \(2\) defective items in a shipment of \(200\) items.

A local bakery produces cookies where the number of chocolate chips in each cookie follows a Poisson distribution with a mean of 3.5 chips per cookie.

(a) Find the probability that a randomly selected cookie contains at least 2 chocolate chips. (2 marks)

(b) A cookie is classified as "premium" if it contains more than 4 chocolate chips. Find the probability that a randomly selected cookie is "premium". (1 mark)

(c) Suppose a box is packed with 6 randomly selected cookies. Find the probability that exactly 2 of these cookies are "premium". (2 marks)