AI-generated synthetic data, in addition to protecting the privacy of original data sets, allows users and data consumers to tailor data to their needs. This paper explores the creation of synthetic data that embodies Fairness by Design, focusing on the statistical parity fairness definition. By equalizing the learned target probability distributions of the synthetic data generator across sensitive attributes, a downstream model trained on such synthetic data provides fair predictions across all thresholds, that is, strong fair predictions even when inferring from biased, original data. This fairness adjustment can be either directly integrated into the sampling process of a synthetic generator or added as a post-processing step. The flexibility allows data consumers to create fair synthetic data and fine-tune the trade-off between accuracy and fairness without any previous assumptions on the data or re-training the synthetic data generator.