  We prove that the Poisson-Boolean percolation on $\mathbb{R}^d$ undergoes a
sharp phase transition in any dimension under the assumption that the radius
distribution has a $5d-3$ finite moment (in particular we do not assume that
the distribution is bounded). More precisely, we prove that:
  -In the whole subcritical regime, the expected size of the cluster of the
origin is finite, and furthermore we obtain bounds for the origin to be
connected to distance $n$: when the radius distribution has a finite
exponential moment, the probability decays exponentially fast in $n$, and when
the radius distribution has heavy tails, the probability is equivalent to the
probability that the origin is covered by a ball going to distance $n$.
  - In the supercritical regime, it is proved that the probability of the
origin being connected to infinity satisfies a mean-field lower bound.
  The same proof carries on to conclude that the vacant set of Poisson-Boolean
percolation on $\mathbb{R}^d$ undergoes a sharp phase transition. This paper
belongs to a series of papers using the theory of randomized algorithms to
prove sharpness of phase transitions.
