If A and B are two events such that \(\rm P(A \cup B) = \frac {3}{4},\;P(A \cap B) = \frac {1}{4},\; P(\bar{A}) = \frac 2 3\) where A̅ is the complement of A, then what is P(B) equal to?

Concept:

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 (A̅) = 1 - P (A) ⇒ P (A) = 1 - P (A̅)

Formula: P (A ∪ B) =  P (A) + P (B) - P (A ∩ B)

Calculation: 

Given: 

\(\rm P(A \cup B) = \frac {3}{4},\;P(A \cap B) = \frac {1}{4},\; P(̅{A}) = \frac 2 3\)

To find: P (B)

As we know, P (A ∪ B) =  P (A) + P (B) - P (A ∩ B)

⇒ P (A ∪ B) =  1 - P (A̅) + P (B) - P (A ∩ B)

\(\Rightarrow \frac{3}{4} = 1-\frac{2}{3} + \rm P(B) - \frac{1}{4}\\\Rightarrow \frac{3}{4} +\frac{1}{4} = \frac{1}{3} + \rm P (B) \\ \therefore \rm P (B) = 1 -\frac{1}{3} =\frac{2}{3} \)