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Agree with you, this quote is confusing. Probability is bounded between zero and one, probability density is non-negative, so "positive probability" taken literally means non-zero. My guess would be that by "positive probability" they mean something like "high probability" and state the tautology "for $x$ such that $p(x)$ is high, the probability of observing $x$ is high". Otherwise they would be suggesting that probability of observing some $x$ value does not correspond to $p(x)$, what is nonsense.