Your testing the probability of the dinosaur dodging 5 times in a row (or the player not hitting 5 times in a row). 1/1024 is the probability of the dinosaur not dodging at all and getting hit 5 times in a row.
The total of an event happening plus the event not happening will always equal 1. The probability of a player dodging 5 times in a row is 243/1024 so the probability of them getting hit on the 5th attempt is 1 - (243/1024) or 781/1024.
Edit: a-ha… just went back and re-read the original post.
Yes, 1/1024 is the correct answer to the situation the OP asked about. As we should be testing for 5 hits in a row or 0.25^5. My apologies to misinterpreting the original question. There is less than a 1% probability that the 5th hit will land.