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Total probability formula
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Sometimes it is useful to split a random experiment in two or more disjoint events whose union is the sample space and use the conditional probabilities with respect to these cases.
For arbitrary events 
and 
in a probability space the following formula holds
To see this, note that 
with disjoint union, and therefore
.
The formula above can be generalized to more than two cases. Suppose that we have a partition of the sample space 
by events 
which means that 
for 
and 
then
.
The latter formula is sometimes called formula of total probability.
Example 7. An urn contains 2 black and 5 white balls. Two balls are drawn without replacement. What is the probability to get a black ball in the second draw.
Solution. Let 
``The second ball is black’’ and 
``The first ball is black’’
We have 
and 
Applying the formula we thus get 
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