Question

Check whether the following probabilities P(A) and P(B) are consistently defined
(i) $P\left(A\right)=0.5,P\left(B\right)=0.7,P(A\cap B)=0.6$
(ii) $P\left(A\right)=0.5,P\left(B\right)=0.4,P(A\cup B)=0.8$

Answer 1

$\text{(i) Here P(A) = 0.5, P(B) = 0.7}$
$\text{and}$ $P(A\cap B)=0.6$
Now $P(A\cap B)>P\left(A\right)$
$\text{Thus the given probabilities are not consistently defined.}$
$\text{Here}P\left(A\right)=0.5,P\left(B\right)=0.4\text{and}P(A\cup B)=0.8\text{We know that}P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)\therefore 0.8=0.5+0.4-P(A\cap B)\Rightarrow P(A\cap B)=0.9-0.8=0.1\therefore P(A\cap B)<P\left(A\right)\text{and}P(A\cap B)<P\left(B\right)$
$\text{Hence, P(A) and P(B) are consistently defined.}$