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Suppose the Republicans won the extremely close 1962 Minnesota gubernatorial race. https://en.wikipedia.org/wiki/Minnesota_gubernatorial_election,_1962 The governor was for the first time elected for four years rather than two. This means that if LBJ were to choose as his running mate one of the two senators from Minnesota--either Hubert Humphrey (who was of course chosen in OTL) or Eugene McCarthy (who seems to have been considered for a while)--and the Democratic national ticket won, it would be Republican governor Elmer Andersen who would decide on Humphrey's or McCarthy's replacement in the Senate until the next election.
Would this fact dissuade LBJ from choosing Humphrey or McCarthy as his running mate? You might say he had more important considerations than a single seat in the Senate. But whereas in the House, a single vote very rarely makes a difference, it can do so in the Senate...