Question

If A and B are two events, then the probability of occurrence of A only is equal to _____________.

Answer 1

For two events A and B, probability of occurrence of A only is $P\left(A\cap \overline{B}\right)$
By Venn-diagram

$$\begin{gathered} \mathrm{Since}A=\left(A\cap B\right)\cup \left(A\cap \overline{B}\right) \\ \therefore P\left(A\right)=P\left(A\cap B\right)+P\left(A\cap \overline{B}\right)-P\left(\left(A\cap B\right)\cap \left(A\cap \overline{B}\right)\right) \\ =P\left(A\cap B\right)+P\left(A\cap \overline{B}\right)-P\left(A\cap B\cap \overline{B}\right)\left[\because \mathrm{Intersection}\mathrm{is}\mathrm{commutative}\right] \\ =P\left(A\cap B\right)+P\left(A\cap \overline{B}\right)-P\left(\mathrm{\varphi }\right)\left[\because B\cap \overline{B}=\varphi \right] \\ P\left(A\right)=P\left(A\cap B\right)+P\left(A\cap \overline{B}\right)-0 \\ \mathrm{i}.\mathrm{e}P\left(A\cap \overline{B}\right)=P\left(A\right)-P\left(A\cap B\right) \end{gathered}$$
∴ Probability of occurrence of A only is P(A) – P(A∩B).