We introduce a novel method to estimate the causal effects of an intervention over multiple treated units by combining the techniques of probabilistic forecasting with global forecasting methods using deep learning (DL) models. Considering the counterfactual and synthetic approach for policy evaluation, we recast the causal effect estimation problem as a counterfactual prediction outcome of the treated units in the absence of the treatment. Nevertheless, in contrast to estimating only the counterfactual time series outcome, our work differs from conventional methods by proposing to estimate the counterfactual time series probability distribution based on the past preintervention set of treated and untreated time series. We rely on time series properties and forecasting methods, with shared parameters, applied to stacked univariate time series for causal identification. This article presents DeepProbCP, a framework for producing accurate quantile probabilistic forecasts for the counterfactual outcome, based on training a global autoregressive recurrent neural network model with conditional quantile functions on a large set of related time series. The output of the proposed method is the counterfactual outcome as the spline-based representation of the counterfactual distribution. We demonstrate how this probabilistic methodology added to the global DL technique to forecast the counterfactual trend and distribution outcomes overcomes many challenges faced by the baseline approaches to the policy evaluation problem. Oftentimes, some target interventions affect only the tails or the variance of the treated units' distribution rather than the mean or median, which is usual for skewed or heavy-tailed distributions. Under this scenario, the classical causal effect models based on counterfactual predictions are not capable of accurately capturing or even seeing policy effects. By means of empirical evaluations of synthetic and real-world datasets, we show that our framework delivers more accurate forecasts than the state-of-the-art models, depicting, in which quantiles, the intervention most affected the treated units, unlike the conventional counterfactual inference methods based on nonprobabilistic approaches.