Question

Two students Anil and Ashima appeared in an examination. The probability that Anil will qualify the examination is 0.05 and that Ashima will qualify the examination is 0.10. The probability that both will qualify the examination, is 0.02. Find the probability that

(i) both Anil and Ashima will not qualify the exam.

(ii) atleast one of them will not qualify the exam.

(iii) only one of them will qualify the exam.

Answer 1

Let A and B denote the events that Anil and Ashima will qualify the examination.

Then, $P\left(A\right)=0.05,P\left(B\right)=0.10$ and $P(A\cap B)=0.02$

(i) Required probability

= P (Anil and Ashima will not qualify the exam)

= $P(A\cup B{)}^{\prime }$

= $1-P(A\cup B)$

= $1-\left[P\right(A)+P(B)-P(A\cap B\left)\right]$

= $1-[0.05+0.10-0.02]$

= $1-0.13=0.87$

(ii) Required probability

= P (atleast one of them will not qualify the exam)

= 1 - P(both of them will qualify the exam)

= $1-P(A\cap B)$

= 1 - 0.02 = 0.98

(iii) Required probability

= P (only one of them will qualify the exam)

= $P(A\cap B^{\prime })+P(A^{\prime }\cap B)$

= $P\left(A\right)-P(A\cap B)+P\left(B\right)-P(A\cap B)$

$[\because P(A\cap B^{\prime })=P(A)-P(A\cap B\left)andP\right(A^{\prime }\cap B)=P(B)-P(A\cap B\left)\right]$

= $P\left(A\right)+P\left(B\right)-2P(A\cap B)$

= $0.05+0.10-2\left(0.02\right)$

= $0.15-0.04=0.11$