The probability of finding a particle described by the normalized state vector |Ψ> in a volume d3r about r is given by
dP(r,t) = |<r|Ψ>|2d3r  =  |Ψ(r,t)|2d3r   (postulate 4).
We may interpret |<r|Ψ>|2 =  |Ψ(r,t)|2 as the probability density.
|<r|Ψ>|2 = <Ψ|r><r|Ψ> is the average value of the observable |r><r|, which we may call the probability density operator.  The probability density operator is the projector onto the ket |r>.  The total probability of finding the particle at time t anywhere in space is
all spaceP(r,t)d3r = ∫all space Ψ*(r,t)Ψ(r,t)d3r = <Ψ|Ψ> = 1.
We expect conservation of probability, i.e.  ∂<Ψ|Ψ>/∂t  = 0.
Let us verify this.
∂<Ψ|Ψ>/∂t = (∂<Ψ|/∂t )|Ψ> + <Ψ|(∂|Ψ>/∂t )
= -(iħ)-1<Ψ|H|Ψ> + (iħ)-1<Ψ|H|Ψ> = 0.
We use iħ∂|Ψ>/∂t = H|Ψ> and -iħ∂<Ψ|/∂t = <Ψ|H,
since H  = H.  Conservation of probability is thus verified.

Is probability locally conserved?
Local conservation of a classical quantity is usually expressed in the following way:
-(∂/∂t)∫Vρ(r,t) dV =∫surface j(r,t)∙n dA = ∫Vj(r,t) dV.
Here ρ(r,t) is the volume density and j(r,t) = ρ(r,t)v(r,t).
The above equation holds for all V.
Therefore
-(∂/∂t)ρ(r,t) =j(r,t).
[ρ(r,t) may stand for particle density or charge density etc, and j(r,t) may stand for particle current density or electrical current density etc.]
Going from classical to quantum physics, we replace expressions for classical quantities which are functions of r and p by operators which are functions of R and P and suitably symmetrize the resulting equations.
ρ --> |r><r|, j --> ρv = ρp/m = (|r><r|P + P|r><r|)/(2m)
We then expect local conservation of probability to translate into
-(∂/∂t)<Ψ|r><r|Ψ>  = ∙(<Ψ|r><r|P + P|r><r|Ψ>)/(2m).
where <Ψ|r><r|Ψ> is the average (expectation) value of the probability density operator and
(<Ψ|r><r|P + P|r><r|Ψ>)/(2m)
is the average (expectation) value of the probability current density operator.
If probability is locally conserved, then
-(∂/∂t)[Ψ*(r,t)Ψ(r,t)] = -(∂/∂t)|Ψ(r,t)|2
= (ħ/(2mi)) ∙[Ψ*(r,t)Ψ(r,t) - (∇(Ψ*(r,t))Ψ(r,t)]
= (ħ/(2mi)) ∙[Ψ*(r,t)Ψ(r,t) - Ψ(r,t)Ψ*(r,t)].
Then
j(r,t) = (ħ/(2mi))[Ψ*(r,t)Ψ(r,t) - Ψ(r,t)Ψ*(r,t)],
j(r,t) =  (1/m) Re[Ψ*(r,t) (ħ/i)Ψ(r,t)]
=  probability current density.

We can verify that this expression is a consequence of the Schroedinger equation by multiplying the Schroedinger equation in the {|r>} representation by Ψ*(r,t) and its complex conjugate by -Ψ(r,t) and adding the two equations.  The above expression is then obtained.  The Schroedinger equation therefore implies local conservation of probability.