Question

If P (A) = 1/3

, P (B^c) = 1/4

, then P (AB) = 0.

True or False?

Answer 1

$False:\\\varOmega =\left[ 0,1 \right] ,P=\lambda \\A=\left[ 0,\frac{1}{3} \right] ,B=\left[ 0,\frac{3}{4} \right] \\P\left( A \right) =\frac{1}{3}\\P\left( B^c \right) =1-\frac{3}{4}=\frac{1}{4}\\P\left( A\cap B \right) =P\left( \left[ 0,\frac{1}{3} \right] \right) =\frac{1}{3}\ne 0$