Question

If A and B are two events such that $P\left(A\cup B\right)=\left(A\cap B\right),$ then P (exactly one of A and B occurs) = _____________.

Answer 1

For two events, $P\left(A\cup B\right)=\left(A\cap B\right),$ then P (exactly one of A and B occurs)

$$\begin{gathered} =P\left(\overline{A}\cap B\right)+P\left(A\cap \overline{B}\right) \\ =P\left(B\right)-P\left(A\cap B\right)+P\left(A\right)-P\left(A\cap B\right) \\ =P\left(A\right)+P\left(B\right)-2P\left(A\cap B\right) \\ \mathrm{Since}P\left(A\cup B\right)=P\left(A\cap B\right)\therefore P\left(A\right)+P\left(B\right)=2P\left(A\cap B\right) \end{gathered}$$

∴ P(exactly are of A and B occurs) = 0