A Study of Adavosertib (MK-1775) in Combination With Paclitaxel and Carboplatin Versus Paclitaxel and Carboplatin Alone for Participants With Platinum-Sensitive Ovarian Tumors With the P53 Gene Mutation (MK-1775-004)

Inclusion Criteria:

* Histologically confirmed non-low grade, non-borderline (low malignant potential) ovarian, fallopian tube, or primary peritoneal cancer which has progressed after paclitaxel / platinum-based therapy.
* Platinum-sensitive disease. Radiological progression must have occurred 6 months or more after the completion of the most recent platinum-based treatment.
* Measurable disease.
* Available tumor sample(s).
* Performance status of ≤1 on the Eastern Cooperative Oncology Group (ECOG) Performance Scale.
* Adequate organ function.

Exclusion Criteria:

* Pregnancy or the intention to become pregnant during the course of the study.
* Participation in a study with an investigational compound or device within 28 days of receiving first dose of study medication.
* Active central nervous system (CNS) metastases and/or carcinomatous meningitis.
* Primary CNS tumor.
* Known hypersensitivity or contraindications to the components of potential study therapy (paclitaxel, carboplatin, adavosertib) or its analogs (i.e., cremophor, mannitol, etc.).
* Participant requires the use of medications or products that are metabolized by, or inhibit, or induce Cytochrome P450 3A (CYP3A4).
* Ongoing peripheral neuropathies ≥Grade 2 and related to previous treatment.
* Known psychiatric or substance abuse disorders.
* Regular use (including "recreational use") of any illicit drugs or recent history (within the last year) of drug or alcohol abuse.
* HIV positive.
* Active Hepatitis B or C.
* Symptomatic ascites or pleural effusion.
* Clinical history suggestive of Li Fraumeni Syndrome.