Mean and variance of a random-guess quiz score

Question

A student guesses on all 40 true/false questions of a quiz, getting each one right independently with probability p = 0.5. Let C be the number of correct answers. What are the mean and variance of C?

Accepted Answer

Counting independent right/wrong outcomes with the same success chance gives a binomial: C \sim 	ext{Binomial}(n=40,\ p=0.5). For any binomial, mean is np and variance is np(1-p): \mathbb{E}[C] = 40 	imes 0.5 = 20, \qquad \operatorname{Var}(C) = 40 	imes 0.5 	imes 0.5 = 10. Standard deviation \sqrt{10} \approx 3.16. A blind guesser scores about 20 out of 40, wobbling by roughly 3 either way. Takeaway. Binomial mean np and variance np(1-p) give 20 and 10 for 40 fair coin-flip questions.

Mean and variance of a random-guess quiz score

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