In a twenty question true false examination a student must achieve eight correct answers to pass. If student answer randomly what is the probability that student will fail.
Let X be the random variable number of correct answers guessed by student.
X ~ Binomial(n = 20, p = 1/2).
Calculate Pr(X < 8).
The slightly more interesting question would be to ask what the pass mark should be if you want the probability of passing purely by guessing each answer to be less than 0.05, say.
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This question reminds me of a true-false examination I was invigilating a couple of years ago. I saw a student tossing a coin for each question (there were 100 questions) and then writing down an answer. When there was 5 minutes remaining I announced this fact to the exam candidates, after which i saw the student frantically tossing the coin as quickly as possible. I went over and asked what was going on. The student informed me that she was checking her answers.