Answer the following question with steps, thank you

Due to COVID restrictions, final exams are currently administered online. For exams that do not use invigilation software there has been an increase in reports of potential collusion between students when sitting the online exam. Identical answers or matching submission times for each question raise red flags for a potential breach of academic integrity. Consider an online exam with 20 questions and answers are directly entered into input boxes on Wattle (that is, no file uploads are permitted). Two students, say Student A and Student B provided an answer to all 20 questions. Suppose for 15 of the 20 questions, the answers from Student A are identical to the answers from Student B, and for the remaining 5 questions, the answers are very similar. As further evidence of collusion, we check the time stamps of when each of the 20 answers were submitted. Let yi = 1 if there is a match between the timestamp of Question i for Student A and the timestamp of Question i for Student B, (2 = 1, ..., 20). Let 0 be the common probability of a match in timestamps for any question. The assumed likelihood function is y:|0 ~ Bern(0). Therefore, if n = 20 is fixed, the joint likelihood is p(y1, ..., y20|0) oc 02:-13: (1-0)"-21-1 : Suppose the observed data are, in order, 1,1,1,0,1,1, 1, 1, 1,1, 1, 1, 1,0,1, 1,0,1,1,1. What criteria should we use to establish a case for collusion between Student A and Student B based on the observed timestamps? Suppose the protocol for measurement is to stop once 15 ones have appeared. (a) [2 marks] Assume a uniform prior on 0. What is the posterior distribution p(@ly) under the new protocol where n is not fixed? (b) [6 marks] Let's run some posterior predictive checks. Define the test quantity T= number of switches between 0 and 1 in the sequence. Simulate the replications yrep under the new measurement protocol to stop once 15 ones have appeared. Display the predictive simulations T(yrep) in a histogram. Compare to the distribution of T(yrep) when n = 20 is fixed and explain any differences