Question

For two events, G and Q, it is given that, P(G) = 2/5, P(Q) = 3/8 andP(G|Q) = 2/3 . If  G̅ and Q̅  are the complementary events of G and Q, then what is \(\rm P(\frac{\bar G}{\bar Q} )\)equal to?
1. 31/25
2. 8/5
3. 21/40
4. 19/25

Answer 1

Correct Answer - Option 4 : 19/25

Concept:

P(A/B) = P(A∩B) / P(B)

\(\rm \overline{ P(A ∪B)}\)=\(\rm { P(\bar A ∩ \bar B)}\rm= 1-{ P(A ∪ B)}\)

P(A ∪ B) = P(A) +P(B) - P(A ∪ B)

 

Calculation: 

Here, P(G) = 2/5, P(Q) = 3/8 and P(G|Q) = 2/3

P(G|Q) = P(G ∩ Q) / P(Q) = 2/3

⇒ P(G ∩ Q) = 2/3 × 3/8 = 1/4

\(\rm P(\frac{\bar G}{\bar Q} )=\frac{P(\bar G∩ \bar Q)}{P(\bar Q)}=\frac{\overline {P( G∪ Q) }}{P(\bar Q)}\)

\(\rm =\frac{1-P(G∪ Q)}{P(\bar Q)}\)

P(Q̅) = 1 - P(Q) = 1-3/8 = 5/8

P(G ∪ Q) = P(G) + P(Q) - P(G∩Q)

= 2/5 + 3/8 - 1/4

= 21/40

\(\rm P(\frac{\bar G}{\bar Q} )=\frac{1-\frac {21}{40}}{\frac 58}\)

= 19/40 × 8/5

= 19/25

Hence, option (4) is correct.