tunneling probability
The tunnelling probability is the chance that a particle striking a barrier it cannot classically cross will nonetheless be found on the far side. It is a number between zero and one, and for a single try at a substantial barrier it is usually very small. Because a particle either tunnels or it does not, the probability is best read as the fraction of a great many attempts that succeed.
Its most important feature is how steeply it shrinks with the size of the barrier. The probability falls off exponentially with the barrier's width and with the square root of how far the barrier's height exceeds the particle's energy. This means doubling the width does not halve the chance — it can slash it by a factor of thousands. A barrier just a few atoms thicker can turn a steady current of tunnelling particles into essentially none.
This savage sensitivity is precisely what makes tunnelling so useful as a probe and so important in nature. A scanning tunnelling microscope reads the spacing between a sharp tip and a surface from the tunnelling current, resolving single atoms because that current changes so violently with distance. In radioactive decay the same steepness explains why half-lives span from fractions of a second to billions of years: small differences in barrier shape produce astronomically different escape rates.
The chance of crossing falls off exponentially with barrier width, so small width changes matter hugely.
A small tunnelling probability per attempt becomes a near-certainty given enough attempts. A trapped particle rattling against a barrier billions of times a second will eventually escape, which is how slow but inevitable processes like radioactive decay arise.