The United States Supreme Court receives roughly 10,000 petitions per year. In order to sift through these petitions and decided what cases to take, the Justices of the Supreme Court first use the "Rule of Four" to decide if a case if worth pursuing. This rule requires four of the nine Justices to vote to formally review the case. If this requirement is met, the Supreme Court issues an order, known as a writ of certiorari, to receive all of the records of the case from the lower court. The Court then formally reviews the case and decides if it will formally accept or deny the petition.
The vast majority of petitions are denied, with only about 80 cases formally heard by the Supreme Court per year. The Supreme Court tends to hear cases that will have large-scale implications, rather than cases that will only affect an individual or a small group of people.
Once the Court has decided to take a case, the case is placed on the court docket, and petitioners and respondents must send in briefs on the case, which can not exceed fifty pages, for the judges to review. The petitioner and respondent are then allowed to respond with shorter briefs to each other's initial brief. Additionally, the Court may allow for interested outside parties to file an amicus curiae brief in which they can express their own arguments and recommendations on the case.
The Court then hears the oral arguments of both parties' lawyers. Each lawyer is given thirty minutes to make their argument and answer questions from the Justices. After the oral arguments are presented by both parties, the Justices are then required to make a decision on the case. They decide the case in what is known as the "Justice's Conference" in the Conference room. As per the protocol of the Supreme Court, the Justices state their thoughts on the case and present any lingering questions of the case to the other Justices. Per protocol, the Chief Justice speaks first, and the rest of the Justices follow in descending order of seniority. Afterward, the Chief Justice casts the first vote on the case, and the remaining Justices casts their votes, in descending order of seniority.
Once the votes are cast and counted, the most senior Justice in the majority assigns a Justice to write the opinion of the Court. Likewise, the most senior Justice in the dissent can assign a dissenting Justice to write a dissenting opinion. Justices then read the concurring and dissenting opinions of the Court and can choose to switch their votes before the official opinion of the Court is released in open court.