The likelihood or actually the UNlikelihood of a satisfactory inflation episode---assuming various beyond-standard-model pictures of cosmology---has recently become a major issue. The discussion revolves around the difficulty of putting a MEASURE on the range of possible initial conditions at the start of expansion. And in models involving a BOUNCE there is the issue of ENTROPY. How can the entropy of the gravitational field (the geometry of the universe) be defined? Assuming a satisfactory definition of entropy, what role might be played by the Second Law of Thermodynamics? Do observers before and after the bounce has different perspectives on the states of the universe and apply fundamentally different coarse-graining, and so on? A conference on Challenges for Early Universe Cosmology was recently held at Perimeter and the talks wrestled again and again with the topics of Measure, Bounce, Geometric Entropy, Probability of Inflation. At the end of the first talk of the conference (by Turok 12 July) a very interesting point was raised by someone in the audience (at time 1:07:40) who pointed out that Loop cosmology addresses these issue in a comaratively simple way. They referred to this paper of Ashtekar and Sloan: http://arxiv.org/abs/1103.2475 Probability of Inflation in Loop Quantum Cosmology Abhay Ashtekar, David Sloan 34 pages, 3 figures Inflationary models of the early universe provide a natural mechanism for the formation of large scale structure. This success brings to forefront the question of naturalness: Does a sufficiently long slow roll inflation occur generically or does it require a careful fine tuning of initial parameters? In recent years there has been considerable controversy on this issue. In particular, for a quadratic potential, Kofman, Linde and Mukhanov have argued that the probability of inflation with at least 65 e-foldings is close to one, while Gibbons and Turok have argued that this probability is suppressed by a factor of ~ 10-85. We first clarify that such dramatically different predictions can arise because the required measure on the space of solutions is intrinsically ambiguous in general relativity. We then show that this ambiguity can be naturally resolved in loop quantum cosmology (LQC) because the big bang is replaced by a big bounce and the bounce surface can be used to introduce the structure necessary to specify a satisfactory measure. The second goal of the paper is to present a detailed analysis of the inflationary dynamics of LQC using analytical and numerical methods. By combining this information with the measure on the space of solutions, we address a sharper question than those investigated in the literature: What is the probability of a sufficiently long slow roll inflation WHICH IS COMPATIBLE WITH THE SEVEN YEAR WMAP DATA? We show that the probability is very close to 1. The material is so organized that cosmologists who may be more interested in the inflationary dynamics in LQC than in the subtleties associated with measures can skip that material without loss of continuity. ========= One of the main points of this paper is that the Loop bounce is simple enough that the universe forms a spacelike hypersurface at the moment of the bounce---making it straightforward to define a probability measure on the range of initial conditions. Models suffering from a singularity at the start of expansion, or where something more elaborate happens (which may involve more complicated assumptions) seem to have a harder time establishing a plausible measure on the initial conditions. We saw a lot of that in the talks at the "Challenges" conference. With some models one had to put a measure not on initial conditions at bounce, but on hypothetical limiting states in the far distant future. Here are videos of all the conference talks: http://pirsa.org/C11008 The opening one, by Turok, at the top of this iist is the one that had the interesting comment towards the end (at time 1:07:40) referring to the Ashtekar et al result in arxiv 1103.2475.