Correct Answer – Option 4 : 0.6

Concept:

The complement of an event:

The complement of an event is the subset of outcomes in the sample space that are not in the event.

The probability of the complement of an event is one minus the probability of the event.

P (A’) = 1 – P (A)

For two events A and B we have

1. P(A ∪ B) = P(A) + P(B) – P(A ∩ B).

2. De morgan’s Law P (A ∪ B)’ = P (A’ ∩ B’)

3. \(P({A’\over B’})={{P(A’\cap B’)\over P(B’)}}={{P(A\cup B)’\over P(B’)}}\)

Calculation:

Given:

\(P(A) = \dfrac{1}{2}, P(B) = \dfrac{1}{4}, P(A \cap B) = \dfrac{1}{5}\)

we have,

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

\(P(A ∪ B) = {1\over 2}+{1\over4}-{1\over5}={11\over 20}\)

The complement of events:

\(P(B’)=1-P(B)=1-{1\over 4}={3\over 4}\)

\(P(A\cup B)’=1-P(A\cup B)=1-{11\over 20}={9\over 20}\)

Now,

\(P({A’\over B’})={{P(A’\cap B’)\over P(B’)}}={{P(A\cup B)’\over P(B’)}}\)

\(P({A’\over B’})={{{9\over20}\over {3\over 4}}}={9\over 20}\times {4\over 3}={3\over 5}=0.6\)