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

S={1,2,3,4,5,6}
The number is even=A={2, 4, 6}
P(A)=\frac{3}{6}$$=\frac{1}{2}
The number is red=B={1,2,3}
$P(B)=\frac{3}{6}=\frac{1}{2}$
$(A∩B)=[2]$
$P(AB)=P(A\cap B)=\frac{1}{6}$
Now
$P(A).P(B)=\frac{1}{2}×\frac{1}{2}$
$=\frac{1}{4}$
Therefore,
$P(A∩B)≠P(A).P(B),$
i.e., A and B are not independent.