How unlikely is it that particles tunnel (comparatively) very long distances? My understanding of quantum tunneling is that when you're not measuring a particle's position you can't know exactly where it is, just where it might be, and that sometimes it happens to be on the other side of the barrier than when you last measured it, explaining some phenomena like certain types of radiation where particles tunnel out of the nucleus and therefore escape the strong force. I don't have the math ability to calculate the probabilities involved, but is there anything stopping a particle from being several metres away from where it was last measured to be? And if so, can particles tunnel away from fields? To me, that would seem to violate the conservation of energy.
 A: The effect is exponential.  The probability of tunneling into a barrier, in the simplest cases, is $e^{-ax}$, where $x$ is the distance into the barrier and $a$ is an constant based on how hard it is to tunnel (associated with the energy levels of the barrier).
So generally we say they can tunnel anywhere, but practically speaking they don't go very far at all.  Any treatment which is more quantified than that is going to require you to provide a whole lot of information about your particular barrier... and the particles in question... far more than you probably care to deal with.
There's an anecdotal story I heard.  A prisoner runs at the wall of his cell once per second.  There is a finite probability that the prisoner will cease to exist inside the cell, and tunnel outside to freedom.  With enough simplifying assumptions, you can even calculate how likely it is to occur.  The answer turns out to be that it is unlikely that the prisoner will escape by these means before the heat death of the universe.
A: The position uncertainty of a particle is not uniform.  A particle's wave function is more likely to spread into regions where the potential energy is low and the particle is classically allowed.  Very little probability leaks through a classically forbidden region, where the potential energy is very high.
One quantitative consequence of this is that the probability of a particle spontaneously tunneling through a potential energy barrier of width $L$ typically goes as $e^{-aL}$.  The quantity $a$ depends on things like the height of the barrier ($a$ is larger if the barrier is higher, making tunneling more difficult), the mass (heavy particles have less position uncertainty and so don't tunnel as far) and energy of the particle, and the geometry.  So the probability of tunneling a large distance when something is trying to hold it in place is typically very small.  (In Mr Tompkins in Wonderland, by the physicist George Gamow, Mr. Tompkins asks how long it will take for his car to quantum-mechanically tunnel out of his garage, and his physics professor companion writes down "100...00 years.")
