Use the notation 𝑃(𝐴|𝐵) and the formula 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴|𝐵)𝑃(𝐵) to solve problems.

Refer to the scenario below. A wildlife ranger is tracking the nesting behaviour of a coastal bird species. Based on historical data: • 65% of nests are located in dense vegetation (event V) • When a nest is in dense vegetation, there is an 82% chance it contains three or more eggs (event E) • When a nest is not in dense vegetation, there is a 45% chance it contains three or more eggs Using the conditional probability formula, calculate the probability that a randomly selected nest is both in dense vegetation and contains three or more eggs. Express your answer as a decimal correct to three decimal places.

A manufacturing plant produces electronic components. Quality inspectors test components in two stages: Stage 1 checks for physical defects, and Stage 2 checks for electrical faults. From historical records, the probability that a component passes Stage 1 is $0.92$. Given that a component passes Stage 1, the probability it also passes Stage 2 is $0.88$. (a) Find the probability that a randomly selected component passes both stages. (1 mark) (b) Find the probability that a randomly selected component fails at least one stage. (2 marks)

A medical clinic records patient test results for two screening tests. The table shows the probabilities for Test A and Test B. Determine the probability that a randomly selected patient tests positive on both tests.