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RolfAndreassen comments on Open Thread, Jun. 29 - Jul. 5, 2015 - Less Wrong
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You take the probability of A not happening and multiply by the probability of B not happening. That gives you P(not A and not B). Then subtract that from 1. The probability of at least one of two events happening is just one minus the probability of neither happening.
In your example of 23% and 48%, the probability of getting at least one is
1 - (1-0.23)*(1-0.48) = 0.60.
Only if A and B are independent.