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Answer to Question #17602 in Statistics and Probability for Katie Wyss

Question #17602
A quiz has twenty multiple choice questions, with five possible answers to each question. Alex has not studied for the quix and has no idea of the correct answers to each question.

A. what is the probability that he gets none of the questions correct?
B. What is the probability that he gets exactly 16 questions correct?
C.What would we expect his score to be out of 20?
Expert's answer
Assume that Alex will choose a variant of answer with thesame probability 1/5.
Then the probability to get incorrect anser is

A. Hence the probability that he gets none of thequestions correct is equal to
(4/5)^20 =0.011529

B. Notice that the number X of correct answers hasbinomial distribution with parameters n=20, p=1/5 and q=1-p=4/5.
Hence the probability that he gets exactly 16 questionscorrect is

P(X=16) = C_20^16 p^16 q^4
=20*19*18*17/(4*3*2) * (1/5)^16 * (4/5)^4

C. The expected score is the expected value of X and isequal to
E(X) = np = 20*4/5 = 16

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