Two fair dice are thrown.Let ‘A’ denote the event that first die shows an odd number and ‘B’denote the event that the second die shows a...

The dice role is random. So each die consists of a set {1
.. 6}. Knowing what was rolled on one die provides no information on what was rolled on
the second die. Pretend the die A is labeled {1 ..6 } while die B is labeled { A .. B }.
There is no union between these two sets, so the event of rolling the die remains
independent. So the answer to your question is that, yes events A and B
are independent.

You can see this by examining the
conditional probability. Two events are independent
if:

P(A|B) = P(A)

P(A|B) = 1/2
= P(A)

P(B|A) = 1/2 =
P(B)

If A and B were on the same die,
the answer would be different.

The standard definition
of independence is:

Two events A and B are independent if
and only if Pr(A ∩ B) = Pr(A)Pr(B).

A is the set: {1,3,5}
--> P(A) = 1/2

B is the set: {1,2,3,5} -->
P(B) = 2/3

A ∩ B is the set : {1,3,5} --> P(A ∩ B) =
1/2

Pr(A ∩ B)
≠ Pr(A)Pr(B)

Therefore these events are
not independent.