1.6.2 Upper Bound of Probability
Until now, we have discussed that probability is a real number between 0 and 1. Through the axiom of nonnegativity, we have established that the probability of an event can't be lower than 0 but didn't address why it should not exceed 1. So, now let's explore why the probability of an event is less than or equal to 1, i.e.:
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Upper Bound Of Probability Proof
Since an event is a subset of sample space, its probability should always be less than or equal to the probability of the sample space:
(1.5)
By the axiom of normalization, we know:
(1.6)
Substituting 1.6 into 1.5 yields:
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