Use a Bernoulli random variable as a model for two-outcome situations.

A quality-control inspector tests computer chips by checking whether each chip passes or fails a voltage stress test. This process can be modelled using a Bernoulli random variable. If the probability that a chip passes the test is $0.92$, which of the following correctly describes the Bernoulli model for this situation?

A quality control inspector tests electronic components for defects. Historical data shows that 8% of components are defective. The inspector randomly selects one component and records whether it is defective or not. Let \(X = 1\) if the component is defective and \(X = 0\) if it is not defective. (a) Identify the probability distribution of \(X\) and state the value of the parameter. [1 mark] (b) Calculate \(E(X)\), the expected value of \(X\). [1 mark] (c) Calculate \(\text{Var}(X)\), the variance of \(X\), to 4 decimal places. [1 mark]

A medical trial tests a new pain relief treatment. The table below shows the number of patients who experienced relief after one dose. (a) Use the table data to estimate the probability \(p\) that a randomly selected patient experiences relief. (1 mark) (b) Define a Bernoulli random variable \(X\) to model whether a single patient experiences relief, using the probability from part (a). State the distribution of \(X\) and determine \(E(X)\). (2 marks) (c) Use the information from part (b) to calculate \(\text{Var}(X)\). (1 mark)

A medical screening test for a disease has a success rate (detecting the disease when present) of 0.92. For each person tested, let $X = 1$ if the test correctly identifies the disease and $X = 0$ if it fails to detect the disease. If 8 people with the disease are screened, which value represents the probability that exactly 6 tests are successful?

A quality control inspector tests electronic components. Each component has a probability of $0.92$ of passing inspection. Let $X$ be a Bernoulli random variable where $X = 1$ if a component passes and $X = 0$ if it fails. (a) Determine $P(X = 1)$ and $P(X = 0)$. [1 mark] (b) Calculate the expected value $E(X)$ and variance $\text{Var}(X)$ for this Bernoulli random variable. [2 marks]