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Mathematics Question on Probability

A fair coin and an unbiased die are tossed.Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’.Check whether A and B are independent events or not.

Answer

S=(H,1),(H,2),(H,3),(H,4),(H,5),(H,6),(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)S={(H,1),(H,2),(H,3),(H,4),(H,5),(H,6),(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)}
Head appears on the coin=A=(H,1),(H,2),(H,3),(H,4),(H,5),(H,6)=6=A={(H,1),(H,2),(H,3),(H,4),(H,5),(H,6)}=6
P(A)$$=\frac{6}{12}
=12=\frac{1}{2}
3 appears on the die=B=(H,3),(T,3)=B={(H,3),(T,3)}
P(B)=212P(B)=\frac{2}{12}
=16=\frac{1}{6}
Now,
(A∩B)=[(H,3)](A∩B)=[(H,3)]
⇒n(A∩B)=1⇒n(A∩B)=1
∴P(A∩B)$$=\frac{1}{12}
Again, P(A).P(B)=12×16P(A).P(B)=\frac{1}{2}×\frac{1}{6}
=112=\frac{1}{12}
Therefore, P(A∩B)=P(A).P(B)P(A∩B)=P(A).P(B), i.e., events A and B are independent.