Manufacturing ship$-$to$-$ship missiles $($designed to be fired by surface vessels against enemy surface vessels, or near$-$surface targets, such as coastal installations$)$ is a challenge. Explosive payloads must be stable during transport $-$ including withstanding potential shocks like train derailments or truck accidents at highway speeds $-$ then stable during long storage times aboard ship, but still volatile enough when triggered to explode forcefully. This balance between stability and explosiveness makes the creation of 'duds' $($missiles that won't function$)$ more likely, and also makes detecting duds more difficult.

Acme Corporation claims to have an industry$-$best process of dud$-$detection that allows them to supply missiles with a dud rate of no more than $1$ in $2,000.$ You are aboard a destroyer supplied with Acme missiles, and have been tasked with checking your inventory for duds using ship's tools, which have an accuracy rate of $98\%,$ that is$,$ in $98\%$ of cases the tools will detect a dud if present, and in $98\%$ of cases where the missile is good will affirm that. Among the $112$ missiles aboard, you find $2$ that register as duds.