Flipping a coin we get either head or tail can't get both so events are mutually exclusive. i.e P(A and B) =0. but flipping the same coin twice may result in either head or tail and result of flipping a coin twice is independent of what appeared the first time. So we can say events are independent? if YEs
then how come it is possible that events are mutually exclusive and like wise independent too ?
On
Let $H_1$ be the event that the first coin was heads. Let $H_2$ be the event the second coin was heads. Similarly, let $T_1$ be the event that the first coin was tails and $T_2$ the event the second coin was tails.
$H_1$ and $T_1$ are mutually exclusive.
$H_1$ and $T_2$ are independent.
You are confusing the first and second statements with one another. There is no contradiction here. The outcome of the first coin toss is independent of the outcome of the second coin toss, i.e. the outcome of the first flip has no impact on the outcome of the second. The outcome of the first coin toss can not be both possibilities simultaneously, i.e. the event that first coin be heads is mutually exclusive of the event that the first coin is tails.
If two events $A$ and $B$ are mutually exclusive then $$ A \rightarrow \neg B $$ and $$B \rightarrow \neg A$$ In particular $$P(A\land B) = P(A\cap B) = 0$$ If two events are independent, then $$P(A\land B) = P(A \cap B) = P(A)P(B) $$ If two events are independent and mutually exclusive, then, combining these two results:
$$P(A\cap B) = P(A)P(B) = 0 \iff P(A) = 0 \lor P(B) = 0$$ Therefore, $A$ must have zero probability or $B$ must have zero probability. Note that this does not imply, however, that any of these events are impossible.