Lifetime of Positron and its existence Since we know in condensed matter the lifetime of positron is about 10^-10 seconds, how do we collide positrons with electrons? How many positrons were there during the LEP experiment?
I know positrons can be produced in beta decays or when two photons 'collide', but do positrons exist naturally such as electrons and protons? And also the positrons used in LEP were produced via beta decays? If not, how?P
 A: Positrons can be easily produced in pair production reactions, when gamma rays with energy more than 1 MeV interact in the field of a nucleus, so there is no problem in producing them at accelerator sites or even from decay products of reactor cores.
For LEP in particular the positrons were generated by using an electron beam hitting a target that would generate gammas and then e+e- pairs, and the e+ was guided to the beam. 

In the left  diagram the gamma comes in from the left and interacts with the field of a nucleus at lower left with outgoing e+ and e-. In the right diagram the interaction is of the incoming  electron is shown with an offshell gamma interacting with the nucleus again, thus there are three outgoing leptons. Magnetic fields can separate electrons from positrons easily. 
In order to get the flux needed for measuring small crossections the beams were accumulated in rings before being injected for the last acceleration and collisions. The number of particles per bunch ( page 16 in link, beams are bunched) for LEP was 50*10^10, for each electrons and positrons.
