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This article develops a Bayesian approach for estimating panel quantile regression with binary outcomes in the presence of correlated random effects. We construct a working likelihood using an asymmetric Laplace (AL) error distribution and combine it with suitable prior distributions to obtain the complete joint posterior distribution. For posterior inference, we propose two Markov chain Monte Carlo (MCMC) algorithms but prefer the algorithm that exploits the blocking procedure to produce lower autocorrelation in the MCMC draws. We also explain how to use the MCMC draws to calculate the marginal effects, relative risk and odds ratio. The performance of our preferred algorithm is demonstrated in multiple simulation studies and shown to perform extremely well. Furthermore, we implement the proposed framework to study crime recidivism in Quebec, a Canadian Province, using a novel data from the administrative correctional files. Our results suggest that the recently implemented "tough-on-crime" policy of the Canadian government has been largely successful in reducing the probability of repeat offenses in the post-policy period. Besides, our results support existing findings on crime recidivism and offer new insights at various quantiles.