Recall that the probability of an event $\,E\,$
is denoted by
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$\,P(E)\,$.
For any event
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$\,E\,$, $\,0\le P(E)\le 1\,$.
That is, a probability is always a number between $\,0\,$ and $\,1\,$.
For a finite sample space (i.e., an experiment with a finite number of outcomes):
a probability of $\,1\,$ means the event will definitely occur;
a probability of $\,0\,$ means the event will definitely
not occur.
Notice that if the experiment is to randomly choose a real number from the interval
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$\,[0,1]\,$,
then the probability of choosing (say) the number $\,0.5\,$ is zero,
and yet this outcome could occur.
This is why a
finite sample space is needed in the above statement.
If
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$\,S\,$ is the sample space,
then $\,P(S)=1\,$.
That is, there's a $\,100\%\,$ chance that
something will happen.
For any events $\,A\,$ and $\,B\,$:
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$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$
If you were just to add the two probabilities, then the intersection gets added twice;
this extra probability must be subtracted.
Recall that
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$\,\overline{E}\,$ denotes the complement of $\,E\,$.
For any event, either it happens, or it doesn't!
Therefore,
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$\,P(E)+P(\overline{E})=1\,$.
Equivalently,
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$\,P(\overline{E})=1 -P(E)\,$.
Recall that $\,\emptyset \,$ denotes the empty set.
Events $\,A\,$ and $\,B\,$ are said to be
mutually exclusive if
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$\,A\cap B = \emptyset\,$.
That is, two events are mutually exclusive when they have nothing in common
(their intersection is empty).
Since $\,P(\emptyset) = 0\,$, for mutually exclusive events, the above formula is simpler:
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$$\,P(A\cup B)=P(A)+P(B)\,$$