A fair die is thrown two times. We will consider the following events:
$A:$ The die first shows an odd number.
$B: $ The die second shows an odd number.
$C:$ The die shows an odd number in both throws or the die shows an even number in both throws.
Prove that the events $A$, $B$, $C$ are not independent.
The first part of the question was asking for a proof that all pairs of the events are independent, which I easily gave by defining the sample space $\Omega$ and showing that $P(X \cap Y) = P(X)\cdot P(Y)$ for all pairs of events. But how do I go about proving dependence for the three events? Any hint would be much appreciated.
Hint: How does the $P(C)$ change when conditioned on $B \cap C$? That is, given events $A$ and $B$, what is the probability of $C$ happening? How does this compare to the unconditioned probability of $C$ happening?
If these probabilities are not the same, then the events cannot be mutually independent.