Positron losing energy but is not destroyed In August 1932, Anderson photographed a positron originating from cosmic rays as it entered a bubble chamber, passed through a 6mm lead plate (in the process loosing energy, as apparent from the changed radius of its path through the chamber).
My question is, as it passed through the 6mm lead plate it did do some interaction with the lead: it lost energy. Why was the positron not annihilated? In its path it had to pass an estimated 1.6 billion electrons. 
What is the proposed mechanism for it to interact with the lead, but avoid annihilation?
 A: As Jon said in the comments, the mean free path before the electron-positron annihilation is longer than 6 millimeters, or at least not much shorter, and it's simply hard for them to annihilate. Most of the space is empty, the electron and positron are pointlike particles, and they're unlikely to hit each other. The mean free path that indicates the "survival distance" of the positron is proportional to the cross section and it is small.
One may get a quick estimate of the survival time by looking at the positronium. It is a bound state of an electron and a positron – totally analogous to the hydrogen atom but with the positron replacing the proton. They also don't annihilate immediately. The fastest decay channel gives the lifetime $1.244\times 10^{-10}$ seconds which, if multiplied by the speed of light (and the cosmic ray positrons have speeds comparable to the speed of light), gives about $3.7$ centimeters.
(The positronium analogy is OK because the penetrating positron basically forms some positroniums most of the time.)
On the other hand, the losing of the energy is via photons that are emitted and that may have arbitrarily low energies, so it's very easy to emit them. Long before the annihilation takes place (and it is a yes/no big event), lots of low-energy photons are emitted.
At the end, the cross sections (areas determining how hard/easy is to "hit" the target, and therefore the probability of a reaction) should be calculated by Feynman diagrams. All the relevant Feynman diagrams basically have some external electron/positron lines, some photon lines, and they are at the tree level. The annihilation differs by its internal electron line (propagator) that is very far from the "mass shell" because this virtual electron must basically interpolate between the original electron and the original positron, turn its momentum backwards in time. So the $1/(p\cdot gamma-m)$ in the propagator has a large denominator so the fraction (propagator) is much smaller than the propagators of nearly on-shell particles. That's what suppresses the annihilation relatively to the emission of soft enough photons.
It is in no way true that the annihilation happens "first". It is a big event and lots of smaller events are taking place "before" the positron and an electron approach each other closely. Even at a large enough distance, they repel each other a little bit, which makes them accelerate (positrons repel from the nuclei, attract to the electrons, thanks, Anna), and this acceleration would lead to the emission of electromagnetic waves even classically. Because the acceleration is small but nonzero at long distances, the corresponding radiation is of low-frequency, and this is matched in quantum field theory by the omnipresent nonzero probabilities to emit low-energy photons whenever the initial (and final) states contain several charged particles.
A: Likely the dominant process for the positron in the lead plate is simply Bhabha scattering.  This is no surprise.  We must consider the lab energy of the positron.
At center-of-momentum energies less than twice the muon rest mass, the electron and positron annihilation is limited primarily to radiation of multiple photons.  At energies above the eV scale, which is definitely the case for this positron going through 6 mm of lead, the probability of this process occurring is very small.  Typically, the positron will need to scatter until it loses enough energy to annihilate.
For a relevant read, see Prantzos, N., C. Boehm, A. M. Bykov, R. Diehl, K. Ferrière, N. Guessoum, P. Jean, et al. “The 511 keV Emission from Positron Annihilation in the Galaxy.” Reviews of Modern Physics 83, no. 3 (September 29, 2011): 1001–56. doi:10.1103/RevModPhys.83.1001.
