Mathematical Methods · Unit 3 · Discrete random variables · Binomial distributions
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A random variable X represents the number of successes in a Bernoulli experiment with n trials, each with probability of success p and probability of failure q. Which of the following statements must be true for X to follow a binomial distribution?
A quality control manager inspects electrical components on a production line. Each component is independently tested and classified as either defective or non-defective. The probability that any single component is non-defective is $0.95$. The manager tests a sample of $8$ components. Which of the following correctly describes the conditions for this scenario to follow a binomial distribution?
A pharmaceutical company tests a new drug on 8 patients. Historical data shows that without treatment, each patient has a probability of 2/5 of recovering naturally within one week. Assume the drug has no effect and each patient's recovery is independent. (a) Determine the values of n and p for this binomial distribution. [1 mark] (b) Establish an expression for the probability that exactly 4 patients recover within one week. [1 mark] (c) Calculate the probability that exactly 4 patients recover within one week. [1 mark] (d) Calculate the probability that at least 6 patients recover within one week. [2 marks]
A quality control inspector examines electronic components coming off a production line. Historical records show that \(15\%\) of components fail an initial test, and each component's test result is independent of all others. (a) Explain why the number of components that fail the test out of a sample of \(8\) components can be modelled as a binomial random variable. State the values of \(n\) and \(p\). [2 marks] (b) The inspector tests \(8\) components. Determine the probability that at most two components fail the test. [2 marks] (c) In a different batch, the probability that a component passes the test is \(q\). If \(12\) components are tested and the probability of exactly \(3\) failures is equal to the probability of exactly \(9\) failures, determine the value of \(q\). [1 mark]
A quality control inspector examines electronic components produced by a manufacturing process. Historical data shows that \(15\%\) of all components fail the inspection. The inspector randomly selects and tests \(8\) components from a production batch. Each component's test result is independent of all others. (a) Identify the values of \(n\) and \(p\) for the binomial random variable representing the number of components that fail inspection in this sample. [1 mark] (b) Determine the probability that exactly \(3\) components fail inspection. Express your answer correct to four decimal places. [2 marks] (c) The inspector's protocol requires a batch to be rejected if \(4\) or more components fail inspection in the sample of \(8\). Calculate the probability that the batch will be rejected. Express your answer correct to four decimal places. [2 marks]