Question

If $P\left(A\right)=0.25,P\left(B\right)=0.5$ and $P(A\cap B)=0.14$ find probability that neither ${}^{\prime }{A}^{\prime }$ nor ${}^{\prime }{B}^{\prime }$ occurs. Also find $P(A\cap \overline{B})$

Answer 1

We have to find $P(\overline{A}\cap \overline{B})=1-P(A\cup B)$ (by De-Morgan's law)
Also, $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$
putting data we get, $P(\overline{A}\cap \overline{B})=0.39$
The shaded region denotes the simultaneous occurrence of $A$ and $\overline{B}$
Hence $P(A\cap \overline{B})=P\left(A\right)-P(A\cap B)=0.11$