  We provide sharp estimates for the distribution function of a martingale
transform of the indicator function of an event. They are formulated in terms
of Burkholder functions, which are reduced to the already known Bellman
functions for extremal problems on $\mathrm{BMO}$. The reduction implicitly
uses an unexpected phenomenon of automatic concavity for those Bellman
functions: their concavity in some directions implies concavity with respect to
other directions. A similar question for a martingale transform of a bounded
random variable is also considered.
