There are two binary vectors with 0s or 1s as values and the correlation between them is calculated; this is done for 20,000 pairs of vectors. Theoretically, is there a difference between having only 3 of these 20,000 correlations having perfect correlation vs. 15,000 of these 20,000 having perfect correlation? Can the perfect correlations in either of these scenarios be trusted more?

Is there a numerical method to actually test whether or not these perfect correlations are meaningful or trustworthy? For instance, after a certain sample size, could having only a couple perfect correlations out of thousands or millions of correlations be caused solely by chance?