  We study the joint laws of a continuous, uniformly integrable martingale, its
maximum, and its minimum. In particular, we give explicit martingale
inequalities which provide upper and lower bounds on the joint exit
probabilities of a martingale, given its terminal law. Moreover, by
constructing explicit and novel solutions to the Skorokhod embedding problem,
we show that these bounds are tight. Together with previous results of Az\'ema
& Yor, Perkins, Jacka and Cox & Ob{\l}\'oj, this allows us to completely
characterise the upper and lower bounds on all possible exit/no-exit
probabilities, subject to a given terminal law of the martingale. In addition,
we determine some further properties of these bounds, considered as functions
of the maximum and minimum.
