Model and solve problems that involve discrete random variables and associated probabilities, with and without technology.

A quality control inspector examines batches of electronic components. The number of defective components, \(X\), in a randomly selected batch follows the probability distribution given by \(P(X = x) = k(5 - x)\) for \(x = 0, 1, 2, 3, 4\), where \(k\) is a constant. (a) Determine the value of \(k\). (1 mark) (b) Calculate the probability that a randomly selected batch contains at most two defective components. (1 mark) (c) Determine the expected number of defective components per batch. (2 marks)

A game at a school fundraiser involves rolling two fair six-sided dice. The random variable \(X\) represents the absolute difference between the two numbers rolled. For example, if a player rolls 5 and 2, then \(X = 3\). (a) Determine the probability distribution of \(X\), expressing each probability as a simplified fraction. (b) Calculate the expected value of \(X\).

A small business operates a customer loyalty scheme. Customers accumulate points with each purchase and may redeem them for rewards. The business manager has analysed historical data and determined that the number of points, \(X\), redeemed by a randomly selected customer in a given month follows the probability distribution shown in the table below. (a) Determine the value of \(k\). [1 mark] (b) Calculate the expected number of points redeemed per customer per month. [2 marks] The business incurs a cost of $0.12 for every point redeemed. Additionally, there is a fixed monthly administration cost of $85.00 for operating the loyalty scheme. (c) Determine the expected total monthly cost of the loyalty scheme if the business has 240 active customers. [2 marks]

A wildlife researcher monitors the breeding success of a particular bird species. Historical data shows that 35% of eggs laid successfully hatch. A clutch consists of 8 eggs. The researcher wants to model the number of eggs that successfully hatch from a single clutch. Which of the following best describes the appropriate probability model for this scenario?

A manufacturing process produces components that are either defective or non-defective. Quality control samples 15 components at random. Historical data shows that 8% of components are defective. The number of defective components in a sample of 15 is modelled as a binomial random variable. Which of the following correctly represents the probability that exactly 3 components in the sample are defective?