Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 14 / 5 / i / Solution

Let $S$ be the number of red vertices in $W$. It has a binomial distribution with parameters $r$ and $1/2$, so independence givesFor $t=1$, the strict inequality $\log \cosh (1/2)<1/8$ follows by integrating $\tanh u<u$ for $u>0$. Hence the exponential Markov bound yieldsThereforeThis supplies the tail estimate directly, including the strict constant in the requested bound.