Multi-objective Bayesian optimization is a sequential optimization strategy in which an optimizer searches for optimal solutions by maximizing an acquisition function. Most existing acquisition functions assume that objectives are independent, but none of them incorporates the correlations among objectives through an explicit formula for exact computation. This paper proposes a novel acquisition function, namely, correlated probability of improvement (cPoI), for bi-objective optimization problems. The cPoI method builds on the probability of improvement and addresses the correlations between objectives by utilizing 3 distinct approaches to compute the posterior covariance matrix from a multi-task Gaussian process. This paper presents both an explicit formula for exact computation of cPoI and a Monte Carlo method for approximating it. We evaluate the performance of the proposed cPoI against 4 state-of-the-art multi-objective optimization algorithms on 8 artificial benchmarks and 1 real-world problem. Our experimental results demonstrate the effectiveness of cPoI in achieving superior optimization performance.