  Suppose you and your friend both do $n$ tosses of an unfair coin with
probability of heads equal to $\alpha$. What is the behavior of the probability
that you obtain at least $d$ more heads than your friend if you make $r$
additional tosses? We obtain asymptotic and monotonicity/convexity properties
for this competing probability as a function of $n$, and demonstrate surprising
phase transition phenomenons as parameters $ d, r$ and $\alpha$ vary. Our main
tools are integral representations based on Fourier analysis.
