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Unit 01 /
Problem 1:


C probability function properties
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Easy


Assume outcome space $\Omega = {a,b}$ (with the usual event space equal to the power set). Prove that the following function is not a valid probability function.

$P(\emptyset) = 0$

$P(a) = .5$

$P(b) = .4$

$P(\Omega) = 1$

Solution:

The given axiom fails the countable additivity axiom. Let event $A = {a}$ and event $B = {b}$. The set ${A,B}$ has just two elements, so it is countable. By countable additivity, $P( \Omega ) = P(A \cup B) = P(A) + P(B) = .5 + .4 = .9$. But $P(\Omega) = 1$, which is a contradiction. Therefore, $P$ is not a valid probability function.