João claims he can read minds. His friend Felipe asked him to guess a number he was thinking of between 1111 and 7777 (inclusive). João guessed Felipe's number correctly 2222 out of 2222 times. Let's test the hypothesis that João cannot read minds and therefore has a chance of 17\dfrac{1}{7}71​start fraction, 1, divided by, 7, end fraction of guessing the correct number each time versus the alternative that his chance is somehow greater.
Assuming the hypothesis is correct, what is the probability of João guessing correctly 2222 times out of 2222 tries? Round your answer, if necessary, to the nearest tenth of a percent.