Understand the concepts of a discrete random variable and its associat ed probability function, and its use in modelling data.

State the domain of a discrete random variable $X$ that represents the number of heads obtained when a fair coin is tossed four times.

A quality control inspector examines batches of electronic components. Let \(X\) represent the number of defective components found in a randomly selected batch of 20 components. Historical data shows that \(X\) follows the probability distribution given in the table below. (a) Find the value of \(k\). (1 mark) (b) Calculate \(E(X)\), the expected number of defective components per batch. (2 marks) (c) Determine \(P(X \geq 2)\), correct to two decimal places. (1 mark)

A café manager claims that the typical customer orders 2 specialty coffees with very little variation. A researcher models the number of specialty coffees ordered per customer using the discrete random variable \(X\) with the probability function shown in the table below. Use calculations of \(\text{E}(X)\) and \(\text{Var}(X)\) to explain whether the data can or cannot support the manager's claim.

A survey records the number of streaming services subscribed to by each household in a random sample of 200 households. Which statement correctly describes a probability function for the discrete random variable representing the number of services?

A book club sends promotional emails to potential members. Let \(X\) be the discrete random variable representing the number of people who respond positively from a group of four recipients. The probability function for \(X\) is shown in the table below. (a) Show that \(k = 0.15\). [1 mark] (b) Calculate \(P(X \geq 2)\). [1 mark] (c) Determine the expected number of positive responses, \(E(X)\). [2 marks]