It is a short review. for more detail, wiki is a good place.
The hall probe is perpendicular to the B field( pointing up) and have a current I passing through ( going forward ).
Due to the Lorentz force. The positron is moving to right and accumulate. The accumulating charge creates a electric force to the left to against further positron accumulate. The magnetic force will be balanced by the electric force. Due to the electric force, there is associated voltage across the hall probe. This voltage is called hall voltage.
Where e is positron charge, v is speed of positron, B is the magnetic field, is the Hall voltage and d is the distance across the hall probe.
The current I is
Where A is the cross section area of the hall probe, n is density of the positron carrier, v is the positron velocity.
Where V is the volume of the hall probe. But the V n is equal to the total number of positron N.
Which is to say, the hall voltage is proportional to the magnetic field.