A wave packet flies at a barrier taller than its own energy. Classically it always reflects — but the wave doesn't vanish instantly inside the barrier; it decays exponentially, and a fraction of it leaks through to the other side.
Real-world example: the scanning tunneling microscope, alpha decay of nuclei, flash memory in your phone.
Barrier height
2.20 E
Barrier width
1.00
Tunneling probability T
2.72 %
Classical physics predicts
0% — always reflection
T ∝ e^(−2κL), where κ ~ √(V₀−E). Increasing EITHER the barrier's height or width exponentially reduces the chance of passing through — which is why tunneling is only noticeable in the microscopic world.