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I have two questions for Probability theory and hope getting your help.

Question 1: Consider the following two experiments:

$T_1$: "Tossing a fair coin".

$T_2$: "Tossing a fair six-sided die".

Let event $A$ be obtaining heads, and event $B$ be rolling a 6. Can we possibly use the addition rule $$P(A \cup B) =P(A)+P(B)-P(A \cap B)$$ for the above events?

Question 2: We know that, if $A$ and $B$ are mutually exclusive events, then $$P(A \cap B)=0.$$ Do both events $A$ and $B$ consider on a single performance of an experiment?

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Q1: Yes, if you're looking for the probability of obtaining a heads or a 6. Q2: Not sure what you mean – DavidP Feb 15 '14 at 11:27
I mean does the definition of mutually exclusive events necessarily consider on a single performance of an experiment? – user53541 Feb 15 '14 at 11:35
The formula for the probability of a union gives no problem. Formally, we may consider the outcomes to be all ordered pairs $(X, k)$, where $X$ is one of $H$ or $T$, and $k$ is an integer from $1$ to $6$. – André Nicolas Feb 15 '14 at 12:08
In practice you can always take the cartesian product of the two universes as a new universe. The only known cases which this is not possible is in quantum mechanics. – kjetil b halvorsen Feb 15 '14 at 12:29
Thank you all, André and kjetil. According to your arguments, I observe that if $P(A), P(B)>0$ in which $A$ and $B$ are events of separate experiments then $A$ and $B$ are never mutually exclusive. Because they are independent so $P(A \cap B) =P(A)P(B)>0$. Is that right? – user53541 Feb 15 '14 at 14:43

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