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For example, in this sentence by the the time you are done reading you will have already skipped over the double "the".

I have searched this on the internet a little, but I have found nothing that focuses on why this happens and shows actual proof and not just a hunch someone has on why it happens. Would someone explain this and post a link to a trustworthy website that explains it in more detail.

double the illusion

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A study by Rainer et al. (2011) has shown that words are skipped and apparently filled in mentally quite often (in the order of 8 to 30% of times).

Two important factors that increased skipping rates were the length of the word and the predictability of the word due to contextual constraints. Both cases apply on the word 'the', because it is short and highly predictable, being a common adjective and in fact one of the most used words in the English language. In other words, besides being short and predictable, it is also a very common word that has relatively little impact on the meaning of a sentence.

Reference
- Rainer et al., J Exp Psychol Hum Percept Perform (2011); 37(2): 514–28

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