Question

If A and B are two events, then the probability of occurrence of exactly one of A and B is equal to __________.

Answer 1

For two events A and B,

Probability of occurrence of exactly are of A and B is $P\left(A\cap \overline{B}\right)+P\left(A\cap \overline{B}\right)$

From previous two questions,

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

i.e Probability of occurrence of exactly are of A and B is $P\left(A\right)+P\left(B\right)-2P\left(A\cap B\right)$