Baumgartner, Prosser, and Crowell are grading calculus exams on which there is a

Question

Baumgartner, Prosser, and Crowell are grading calculus exams on which there is a True - False question with ten parts. Baumgartner notices that there is a student with only two of the ten parts correct, and remarks, "This student was not even bright enough to have flipped a coin to determine his answers." "Not so clear" says Prosser. "With 340 students I bet that if they all flipped coins to determine their answers there would be at least one exam with two or fewer answers correct." Crowell says, "I'm with Prosser. In fact I bet we should expect at least one exam in which no answer is correct if everyone is just guessing."
What is the probability that there is a least one exam with no answer correct if everyone is guessing?

Explanation / Answer

P(a person gets all wrong answers) = .5^10

P(a person doesn't get all wrong answers) = 1 - .5^10
P(none of 340 get all wrong answers) = (1-.5^10)^340
P(at least 1 of 340 gets all wrong answers) = 1 - (1-.5^10)^340 = 0.2827

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