For two events A and B, let \({\rm{P}}\left( {\rm{A}} \right) = \frac{1}{2},{\rm{\;P}}\left( {{\rm{A}} \cup {\rm{B}}} \right) = \frac{2}{3}\) and \({\rm{P}}\left( {{\rm{A}} \cap {\rm{B}}} \right) = \frac{1}{6}\). What is the P(A̅ ∩ B) equal to?

Concept:

Formulas used:

• P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• P(A̅ ∩ B) = P(B) - P(A ∩ B)

Calculation:

Given: \({\rm{P}}\left( {\rm{A}} \right) = \frac{1}{2},{\rm{\;P}}\left( {{\rm{A}} \cup {\rm{B}}} \right) = \frac{2}{3}\) and \({\rm{P}}\left( {{\rm{A}} \cap {\rm{B}}} \right) = \frac{1}{6}\)

To find: P(A̅ ∩ B)

We know that, P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

So, P(B) = P(A ∪ B) + P(A ∩ B) - P(A)

\(\Rightarrow {\rm{P}}\left( {\rm{B}} \right) = \frac{2}{3} + \frac{1}{6} - \frac{1}{2}\)

⇒ P(B) = 2/6

⇒ P(B) = 1/3

P(A̅ ∩ B) = P(B) - P(A ∩ B)

⇒ P(A̅ ∩ B) = 1/3 - 1/6

⇒ P(A̅ ∩ B) = 1/6