Answer to Question #112326 in Statistics and Probability for Mariamyussifsaeed

Question

28. A candidate is taking a multiple-choice exam. For each tested in the exam, there are
5 possible choices i.e A, B, C, D, and E. Because the candidate zero knowledge of the
subject, he relies on pure guesswork to answer each question independent of how he
answers any previous questions. Find
(a) the probability that the candidate answers three questions wrong in a row before
he finally answers the fourth question correctly.
(b) let X= the number of problems the candidate answers wrong in a row before he
finally guesses a correct answer.

Answer 1

Because the candidate zero knowledge of the subject, and he relies on pure guesswork to answer each question independent of how he answers any previous questions the probability wrong answer is always $P_w=4/5=0.8$. the probability correct answer is $P_c=1/5=0.2$ . The probabilities of independent events are multiplied. Therefore, the probability of giving the first three wrong answers, and then the correct one is $P_{3,1}=P_w^3\cdot P_c=0.8^3\cdot 0.2=0.1024$

It should be explained that exactly the same probability corresponds to any sequence of alternating one correct answer with the wrong three, if we consider an ensemble of four attempts at answers.

Probability X of incorrect answers preceding the correct one is

$P_{X,1}=P_w^X\cdot P_c=0.8^X\cdot 0.2$

Answer: (a) the probability that the candidate answers three questions wrong in a row before

he finally answers the fourth question correctly is $0.1024$ ;

(b) the probability that the candidate answers X=n questions wrong in a row before

he finally answers the question correctly is $0.8^n\cdot 0.2$ .