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Using martingale theory, we compute, in very few lines, exact analyticalexpressions for various first-exit-time statistics associated withone-dimensional biased diffusion. Examples include the distribution for thefirst-exit time from an interval, moments for the first-exit site, andfunctionals of the position, which involve memory and time integration. As akey example, we compute analytically the mean area swept by a biased diffusionuntil it escapes an interval that may be asymmetric and have arbitrary length.The mean area allows us to derive the hitherto unexplored cross-correlationfunction between the first-exit time and the first-exit site, which vanishesonly for exit problems from symmetric intervals. As a colophon, we exploreconnections of our results with gambling, showing that betting on thetime-integrated value of a losing game it is possible to design a strategy thatleads to a net average win.

On the area swept by a biased diffusion till its first-exit time: Martingale approach and gambling opportunities