Binomial distribution：練習問題

A specific breed of flower has seeds that germinate with a probability of \(0.8\). A gardener plants a random sample of \(15\) seeds. Find the probability that exactly \(12\) of these seeds germinate.

State two necessary assumptions required for a random variable to be modelled effectively by a binomial distribution in a real-world context.

For a binomial distribution \(X \sim B(n, p)\), it is given that the mean is 4.8 and the variance is 2.88. Find the values of \(n\) and \(p\).

A company claims that 95% of its deliveries arrive on time. A researcher suspects the claim is too high and tests a random sample of 100 deliveries, finding that 91 were on time. Test the company's claim at the 5% significance level using a binomial model.

A large batch of electronic switches is tested for quality. The probability that any single switch is faulty is 0.04. A random sample of 50 switches is selected for inspection.

(a) Find the probability that exactly two switches in the sample are faulty. (2 points)

(b) Find the probability that at least three switches in the sample are faulty. (2 points)

A company produces glass jars. The probability that a jar is defective is given by \(p\). A random sample of 20 jars is taken. Let \(X\) be the number of defective jars in the sample.

(a) Given that the mean of \(X\) is 1.6, find the value of \(p\). (2 points)

(b) Using the value of \(p\) found in part (a), find the probability that there are at most 3 defective jars in the sample. (2 points)

(c) Calculate the variance of \(X\). (1 point)