Question

A question on a particular exam provides the student with with 5 questions and 5 answers that must be matched. If the student is clueless does the matching at random, find (a) the probability the first question is matched correctly (b) the probability that the first two questions are answered correctly. (c) the probability that at least one of the questions is answered correctly.

Answer 1

$We~can~arrange~5~answers~for~5~questions~in~5!~ways~which~are~mutually\\ ~exclusive,~exhaustive~and~equally~likely.\\ (a)~Let~A~be~an~event~that~the~first~question~is~matched~correctly.~\\Then~n(A)=total~no.~cases~favourable~to~event~A=4!\\ Required~probability=P(A)=\frac{n(A)}{5!}=\frac{1}{5}.\\ (b)~Let~B~be~an~event~that~the~first~two~questions~are~matched~correctly.~\\Then~n(B)=total~no.~cases~favourable~to~event~B=3!\\ Required~probability=P(B)=\frac{n(B)}{5!}=\frac{1}{20}.\\ (c)~Let~A_i~be~an~event~that~ith~question~is~matched.\\ Then~P(A_i)=\frac{4!}{5!}=\frac{1}{5},~i=1(1)5,\\ P(A_i\cap A_j)=\frac{3!}{5!}=\frac{1}{20},~i<j\\ P(A_i\cap A_j\cap A_k)=\frac{2!}{5!}=\frac{1}{60},~i<j<k~and~so~on.\\ Then~required~probability=P(\cup_{i=1}^5A_i)\\ =\sum_{i=1}^5P(A_i)-\sum_{i<j}^5P(A_i\cap A_j)&plus;\sum_{i<j<k}^5P(A_i\cap A_j\cap A_k)-\sum_{i<j<k<l}^5P(A_i\cap A_j\cap A_k\cap A_l)&plus;P(A_1\cap A_2\cap A_3\cap A_4\cap A_5)\\$

$=\binom{5}{1}\times \frac{1}{5}-\binom{5}{2}\times \frac{1}{20}&plus;\binom{5}{3}\times \frac{1}{60}-\binom{5}{4}\times \frac{1}{5!}&plus;\binom{5}{5}\times \frac{1}{5!}\\ =1-\frac{1}{2!}&plus;\frac{1}{3!}-\frac{1}{4!}&plus;\frac{1}{5!}=0.6333$