Answer to Question #110397 in Statistics and Probability for Emmanuel Daniels

Question

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 has 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 finally 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.

(c) When the candidate's exam is being graded, the grader finds that the candidate
has answered the first five exam questions all wrong. Can the grader conclude that
the candidate is more or less likely to guess a problem right in other problems.

Answer 1

 Let X= the number of problems the candidate answers wrong in a row before he finally guesses a correct answer. 

(a)

$$P(X=3)=({4 \over 5})^3({1 \over 5})={64 \over625}=0.1024$$

(b)

$$P(X=x)=({4 \over 5})^x ({1 \over 5})={4^x \over5^{x+1}}, x=0,1,2,...,N-1$$

(c)

Probability that the candidate guess a problem right in other problems is

$$Pr=({1\over 5})^{N-5},$$

$N$ is the total number of problems.

The grader cannot conclude that the candidate is more or less likely to guess a problem right in other problems because the answers of different questions are independent events.