Not much, according to Andrew Gelman:

[Y]ou can’t learn much from primary elections. They can be important in their effectsboth directly on the composition of Congress and indirectly in how they can affect behavior of congressmembers who might be scared of being challenged in future primariesbut I don’t see them as very informative indicators of the general election vote. Primaries are inherently unpredictable and are generally decided by completely different factors, and from completely different electorates, than those that decide general elections.

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