Question

A die marked 1,2,3 in red and 4,5,6 in green is tossed. Let A be the event, 'number is even' and B be the event, 'number is red'. Are A and B independent ?

Answer 1

When a die is thrown, the sample space S={1,2,3,4,5,6}.

Also, A : number is even and B : number is red $\therefore A=(2,4,6),B=(1,2,3)andA\cap B=2.\Rightarrow n\left(A\right)=3,n\left(B\right)=3,n(A\cap B)=1Now,P\left(A\right)=\frac{3}{6}=\frac{1}{2},PA\cap B=\frac{1}{6}andP\left(A\right)\times P\left(B\right)=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}\ne \frac{1}{6}=P(A\cap B)\Rightarrow P(A\cap B)\ne P\left(A\right)P\left(B\right)$

Thus, A and B are not independent events.