Docetaxel and cabazitaxel are cancer chemotherapy agents of the taxane drug class. The purpose of this study is to explore the benefit, for treatment of metastatic castration-resistant prostate cancer (mCRPC), of a regimen in which participants begin treatment with either of these two taxane drugs (docetaxel or cabazitaxel, in combination with prednisone) and are switched to the other taxane drug if prostate-specific antigen (PSA) value does not decrease ≥30% after 4 cycles. As defined in study protocol amendment 3, efficacy results are summarized for all participants combined, irrespective of which agent (docetaxel or cabazitaxel) was administered initially, rather than separately for the two groups based on taxane administered initially. One of the primary outcome measures is percentage of participants with a ≥50% sustained decrease from baseline in PSA at any time during the trial. By providing an opportunity for patients to switch taxane based on early PSA response, there may be a difference in result for this measure versus result in a study where it was not possible to switch. The other primary outcome measures are change from baseline in circulating tumor cells (CTCs) biomarkers percent androgen receptor nuclear localization (%ARNL) and microtubule bundling (MTB).