What is the probability of randomly guessing at least 8 out of 10 questions correct on a true-or-false quiz?

P( at least 8 correct ) = 
P( 8 ) + P( 9 ) + P( 10 ) = 
(10 C 8).5^10 + (10 C 9).5^10 + (10 C 10).5^10 = 56*.5^10 = 0.0546875 

Note: Let C denote correct and I denote incorrect. Suppose that guessing results in 8 correct and 2 incorrect answers. One possible outcome is II, for example. We can think of the number of ways of getting exactly 8 correct to be the same as the number of ways of selecting 8 spots for the correct answers times the number of ways of selecting two spots for the remaining two answers. There are (10 C 8) ways to select the positions for the correct answers and precisely one way to select the remaining two positions for the incorrect answers. Therefore, there are there are 10 C 8 = 45 ways to get 8 correct. Similarly, there are 10 C 9 = 10 ways to get 9 correct, and 10 C 10 = 1 way to get all 10 correct.