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What I seen so far,

The probability of tossing heads is 1/2.
The probability of tossing tails is 1/2.

Therefore, the probability of tossing a coin for either tails or heads is 1 which is a mutually exclusive event.

Is it possible to show an example of non-mutually exclusive event using such a single coin example? If not then what are it's requirements?

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    Yes, but you say :...tossing a coin for either heads or tails... which is **a** mutually exclusive **event.**" (emphasis added). The **single** event "either heads or tails" which you correctly say has probability $1$ cannot be **a** mutually exclusive **event**; as I said in my first comment, **mutual** exclusion requires at least two events. The **two** events "outcome is head" and "outcome is tail" are mutually exclusive events: the **single** event "either heads or tails" cannot be called **a** mutually exclusive event.2012-10-11

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Event A: The coin comes up heads.
Event B: The coin comes up either heads or tails.

The probability of A is $\frac12$; the probability of B is $1$.

For less trivial examples you need more than two possible outcomes. Flipping a coin twice (or flipping two coins) will work, as will rolling a die. With two coin tosses, for instance:

Event A: I get at least one head.
Event B: I get at least one tail.

These are not mutually exclusive: both occur if I get HT or TH. They’re also not identical: if I get HH, A occurs but B doesn’t. Each has probability $\frac34$.

With a die:

Event A: the number that comes up is even.
Event B: The number that comes up is $1,2$, or $3$.

If I roll a $2$, both A and B occur, so they’re not mutually exclusive. Note that in this case each has probability $\frac12$, so their probabilities do add up to $1$, even though they are not mutually exclusive.

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    To add a little to dispel what seems to be another idea embedded in the OP's mind, the events $A$ and the event $C$: "roll a $3$" are mutually exclusive events of probabilities $\frac{1}{2}$ and $\frac{1}{6}$. Thus, P(A\cup C) = \frac{2}{3} < 1 showing that it is _not necessary_ that the union of mutually exclusive events have probability $1$; smaller probabilities are possible. It **is true**, though, that **the probability of the union of mutually exclusive events cannot exceed** $1$.2012-10-11
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According to wikipedia, the toss of a single coin is a clear example of mutual exclusivity (https://www.wikiwand.com/en/Mutual_exclusivity). The outcome is either heads or tails, but not both. That seems to contradict: "The single event "either heads or tails" which you correctly say has probability 1 cannot be a mutually exclusive event; as I said in my first comment, mutual exclusion requires at least two events." I think this first explanation has got it reversed. A single coin flip results in one of two mutually exclusive events. Repeated tosses are independent of one another. Correct?