11. A committee of three is selected uniformly at random from five economists and four engineers. Given that the committee includes at least one engineer, what is the probability that it includes exactly two engineers?
12. An admissions office used an algorithm to identify applicants most likely to miss a document deadline and sent reminders only to the flagged applicants. Their completion rate was lower than that of applicants who received no reminder. A manager concludes that reminders discouraged completion and should be stopped. Which investigation would provide the strongest basis for assessing the reminders' effect, rather than differences between the groups that received and did not receive them?
13. An organisation that pays staff for meeting a measured target may obtain more than the recorded improvement it wants, or less. A repair service rewarded for closing cases quickly may become genuinely more efficient. It may also close difficult cases prematurely or record a reopened case as a new one. These possibilities do not show that targets are useless: without a common measure, managers can struggle to identify persistent delays. They show that a measure's meaning depends partly on the practices it encourages. A safeguard can therefore change the value of the original measure. Counting reopened cases, for example, may make rapid closure more informative, although it can also discourage staff from accepting unusually difficult work. The appropriate question is not simply whether performance rose after a target was introduced, but which activities generated the rise and what the target left unmeasured. Which option best captures the passage's main argument?
14. Positive integers x and y satisfy x > y x y x − y = 6
15. Two panels independently assess the same one hundred research proposals using different published criteria. Panel A recommends sixty proposals for funding; panel B recommends forty. A manager observes that A recommends a greater proportion and therefore calls A the more permissive panel. The manager then concludes that every proposal recommended by B would also be recommended by A, so consulting B could not add a proposal to A's recommended list. What is the central flaw in this reasoning?
