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

A quality control technician tests electronic components produced at a factory. Each component is classified as either defective (failure) or non-defective (success). The probability that a randomly selected component is non-defective is $0.92$. The technician tests components one at a time and stops after the first defective component is found. Let $X$ be a Bernoulli random variable representing the outcome of a single component test, where $X = 1$ denotes success (non-defective) and $X = 0$ denotes failure (defective). (a) State the probability mass function $P(X = x)$ for this Bernoulli random variable. [1 mark] (b) Calculate $E(X)$, the expected value of $X$. [1 mark] (c) Calculate $\text{Var}(X)$, the variance of $X$. [1 mark]

A production facility manufactures electronic components. Historical data shows that each component manufactured has a probability of 0.12 of being faulty. A quality inspector randomly selects one component each hour to test. Let $X$ be a Bernoulli random variable where $X = 1$ if the component is faulty and $X = 0$ if the component is not faulty. (a) Determine the probability that the component selected in the next inspection is not faulty. (1 mark) (b) Calculate the expected value (mean) of $X$. (1 mark) (c) Calculate the variance of $X$. (1 mark)

A technician inspects components on a production line. The probability that any single component is defective is 0.15. If the technician inspects 8 components, which of the following correctly describes the number of defective components as a Bernoulli-related random variable?

A quality control inspector examines items coming off a production line. Historical data shows that each item has a probability of \(0.15\) of being defective, independent of all other items. (a) Model this situation as a Bernoulli random variable \(X\), where \(X = 1\) represents a defective item and \(X = 0\) represents a non-defective item. Determine \(\text{E}(X)\) and \(\text{Var}(X)\). [2 marks] (b) The inspector examines \(8\) items in a random sample. Determine the probability that exactly \(2\) items are defective. Express your answer correct to \(4\) decimal places. [2 marks]