How is set theory related to probability?
We have seen that in general the prior information about the occurrence of events influences the likelihood of other events occurring. So in general:
If but is, so the occurrence depends on not from the occurrence (or non-occurrence) of the event from.
and are then from each other independent events (stochastic independence). Because if is independent of so is too independent of .
Then the following must apply:
Notice the difference between the terms "independent" and "disjoint":
Are two events and With and disjoint, then:
because of for the probability: ;
So that and cannot be independent of one another if they are disjoint, and that if they are independent of one another they must have some common intersection.
Thus disjoint and independent are mutually exclusive.
Deriving relationships in the case of independence
What has to be shown is that when you are independent .
are and independent events, then
Multiply both sides by and multiplying the left side gives:
After adding the second term on the left and applying the definition for surrendered:
In the same way one can show that applies.
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