- #1
phosgene
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Homework Statement
If A and B are events, use the axioms of probability to show that:
[itex]if B \subset A, then P(B) \leq P(A)[/itex]
Homework Equations
Axiom 1: [itex]P(n) \geq 0[/itex]
Axiom 2: [itex]P(S)=1[/itex]
Axiom 3: If A1,A2,... are disjoint sets, then [itex]P(\bigcup _{i} A_{i}) = \sum_{i} P(A_{i})[/itex]
The Attempt at a Solution
I start with using the law of total probability to define the set A:
[itex]A= (A \cap B) \cup (A \cap B^{C})[/itex]
Then I use axiom 3 to get turn it into a probability:
[itex]P(A) = P(A \cap B) + P(A \cap B^{C})[/itex]
Since [itex]B \subset A, P(A \cap B) = P(B)[/itex]
So
[itex]P(A) = P(B) + P(A \cap B^{C})[/itex]
[itex]P(B)=P(A) - P(A \cap B^{C})[/itex]
And as axiom 1 states that a probability must be greater than or equal to 0,
[itex]P(B) \leq P(A)[/itex]
As for proving the equality case, this means that [itex]P(A \cap B^{C}) = 0[/itex], but then doesn't that just mean that A=B. Since the question states that B is a *proper* subset of A, am I incorrect in thinking that it might be a typo?
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