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==History of bounding probability==
The idea of bounding probability has a very long
tradition throughout the history of probability theory. Indeed, in 1854 [[
| last = Boole
| first = George
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positive variable when only the mean is known.
[[Henry E. Kyburg, Jr.|Kyburg]]<ref name="kyburg99">Kyburg, H.E., Jr. (1999). [http://www.sipta.org/documentation/interval_prob/kyburg.pdf Interval valued probabilities]. SIPTA Documention on Imprecise Probability.</ref> reviewed the history
of interval probabilities and traced the development of the critical ideas through the [[20th century]], including the important notion of incomparable probabilities favored by [[John Maynard Keynes|Keynes]].
Of particular note is [[Maurice René Fréchet|Fréchet]]'s derivation in the [[1930s]] of bounds on calculations involving total probabilities without
dependence assumptions. Bounding probabilities has continued to the
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==Logical expressions==
Logical or [[Boolean_function|Boolean expressions]] involving [[logical_conjunction|conjunctions]] ([[AND_gate|AND]] operations), [[logical_disjunction|disjunctions]] ([[OR_gate|OR]] operations), exclusive disjunctions, equivalences, conditionals, etc. arise in the analysis of fault trees and event trees common in risk assessments. If the probabilities of events are characterized by intervals, as suggested by [[George Boole|Boole]]<ref name="BOOLE1854" /> and [[John Maynard Keynes|Keynes]]<ref name="kyburg99" /> among others, these binary operations are straightforward to evaluate. For example, if the probability of an event A is in the interval P(A) = ''a'' = [0.2, 0.25], and the probability of the event B is in P(B) = ''b'' = [0.1, 0.3], then the probability of the [[logical conjunction|conjunction]] is surely in the interval
: P(A & B) = ''a'' × ''b''
:::: = [0.2, 0.25] × [0.1, 0.3]
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