A dark horse
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The primary elections are for delegates to the National nomination convention. Not all states have them and the way in which they select delegates varies. Sometimes the delegation is divided in proportion to the vote, sometimes the one with the most votes get all of the delegates. Sometimes the primaries are only advisory information for the delegates. If no candidate has a majority of the bound and committed delegates by convention time, delegates are all free after the first ballot to vote as they choose. Ballots are taken and deals are made and delegates switch votes until finally one candidate has a majority.
a plurality.
Safe District : Electoral district in which the candidate from the dominant party usually wins by 55 percent or more.
That does happen, and it's not as rare as I thought. Those who do challenge an incumbent President for the party nomination are usually not a serious threat, but there have been a few challenges in the past half century worth mentioning. For example, in 1976, Ronald Reagan competed against incumbent President Gerald Ford for the Republican Party Nomination. That race was too close to call right until the Republican National Convention, where Ford narrowly won the nomination. He lost the election, however. Also, in 1992 Pat Buchanan ran against incumbent President George H. W. Bush for the Republican Nomination. 73% of Republicans voted for Bush in the primaries. In 1980 Ted Kennedy (the U.S. Senator from Mass. who died in 2009 and brother of the former President) competed for the Democratic Nomination against incumbent Jimmy Carter. Although Carter had 24 Primary wins to Kennedy's 10, Kennedy refused to concede until he lost the nomination in a 2129 to 1146 vote at the Convention. Many were surprised when Eugene McCarthy ran against Lyndon Johnson for the 1968 Democratic nomination. Four years earlier Johnson had the highest percentage of popular votes of any U.S. presidential candidate since George Washington. After Johnson received only 49% of the vote at the New Hampshire primaries to McCarthy's 42%, Robert Kennedy also entered the race against Johnson. It became obvious to Johnson that the Democratic nomination was something that he was going to have to work for, but all his time was consumed by the war in Vietnam as well as the urban racial unrest domestically, so he withdrew from the election at the end of March 1968. Pete McCloskey and John Ashbrook challenged Richard Nixon for the 1972 Republican nomination. Out of 1324 delegates to the Republican Convention, Nixon won 1323 and McCloskey won 1.
One example of majority rule is the democratic voting system. The candidate with the most votes gets the delegates in the state and wins the election.