How does NASA determine the position of their probes? I'm currently following New Horizons visit to Pluto. I'm just wondering how does the Probe/NASA determine the Position of the Probe? I would assume using earth's GPS wouldn't have the necessary angular resolution if you're billions of km far away.
They flew billions of km and missed by a mere 70km. 
What is the accuracy of the positioning system?
 A: 
I'm just wondering how does the Probe/NASA determine the Position of the Probe?

There are several things that can be directly measured from Earth, as discussed in detail at https://solarsystem.nasa.gov/basics/bsf13-1.php.  I describe these briefly below.


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*radial velocity toward/away from Earth:  using the Doppler-shift of a well known downlink carrier signal

*distance from Earth: using a ranging pulse encoded into the uplink signal and a known turnaround time within the spacecraft electronics plus other known delays, one can determine the $\Delta t$ (i.e., time delay) of the travel to/from the spacecraft

*angular position in sky:  using Doppler effects from the Deep Space Network (DSN) stations for low-accuracy values and Very-long-baseline interferometry for more accurate measurements


Using multiple ground stations, we can improve the distance/range, position, and velocity estimates.  The uncertainties are generally driven by financial limitations and/or mission necessity (e.g., some spacecraft science objectives do not require that we know their position to better than ~100 km).  Some spacecraft can take advantage of the Global Positioning System (GPS), in the same way one would triangulate their position on Earth, if the spacecraft is below the GPS orbit (i.e., ~20,200 km altitude).  GPS trackers on spacecraft can be expensive and it can sometimes be easier/cheaper to determine attitude information from the Near Earth Network using similar techniques to those described above using the DSN.

What is the accuracy of the positioning system?

This depends upon the mission and goals/requirements for attitude control.  Onboard star and sun sensors provide the spacecraft orientation, generally to less than a degree (sometimes to much smaller tolerances if necessary, e.g., space telescopes often require very high accuracy).  The velocities can be determined down to mm/s (or less if necessary) and the position can be known down to less than one kilometer.  For instance, Cassini's orbital position must be very well known ($\leq1 \ km$) due to the large amount of orbital debris around Saturn.
Other spacecraft in less hazardous environments require less accurate attitude information.  For instance, the Wind spacecraft is currently orbiting the $L_{1}$ point and its position need not be known to better than 10s of km.

They flew billions of km and missed by a mere 70km.

I am not sure from where the distance 70 km arose because New Horizons was roughly ~7800 km from Pluto and ~28,800 km from Charon at closest approach.
