Probability on Trees and Networks, by Russell Lyons and Yuval Peres
Markov Chains and Mixing Times (2nd edition), by David Levin and Yuval Peres with Elizabeth Wilmer
The Probabilistic Method (3rd edition), by Noga Alon and Joel Spencer
Brownian Motion, by Peter Moters and Yuval Peres
Probability: Theory and Examples (4th edition), by Rick Durrett
Aim for the writing style of a research paper. Many scientfic papers end with "acknowledgements'' where you can thank X and Y for inspiring conversations, Z for pointing out a gap in your proof, W for feeding and encouraging you when the going got tough, and your favorite coffee shop for providing an essential raw material.
The vast majority of math papers are typeset using LaTeX.
Toward the end of the semester, you'll present in a group of 2-5 on a probability research topic of your choice. You can choose between two fora:
The length of the presentation is 50 minutes if you present in class, or up to 75 minutes if you present in the arXiv seminar. Your group can choose how to split up the time among the group members. In that time the group should aim to state one theorem, place it in context (why is it interesting? what gap in human knowledge does it fill? what related things are true/false/easy/hard/known/unknown?), and convey the main idea of the proof.
Expect a lot of questions from me and your classmates that will slow you down! If a practice presentation takes 30 minutes with no questions, then you're in good shape.