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If A, B are two events, satisfying the equation
\ ({P \ left (AB \ left) = P \ left (B \ left | A \ left) P \ left (A \ left) = P \ left (A \ left) P \ left (B \
right) \ right. \ right. \ right. \ right. \ right. \ right. \ right. \ right. \ right.} \) called events A, B are independent, referred to as A, B independently -
Independent test was repeated
to a separate test is repeated n times, the probability of event A when each embodiment is p, then the probability of k times n trials is occurring A: \ ({P \ left (X-k = \ left ) = C \ mathop {{} } \ nolimits _ {{n}} ^ {{k}} p \ mathop {{}} \ nolimits ^ {{k}} \ left (1-p \ left) \ mathop {{ }} \ nolimits ^ {{nk }} \ right. \ right. \ right. \ right.} \)
Example: a system consists of four components Ⅰ, Ⅱ, Ⅲ, Ⅳ, and four members are independent, the overall system reliability requirements.
Solution: Write A = {whole system work}
Ai = {a member work}
\ ({P \ left (A \ mathop {{}} \ NoLimits _ {{. 1}} A \ mathop {{}} \ nolimits_ {{2}} \ left) + P \ left (A \ mathop {{}} \ nolimits _ {{3}} A \ mathop {{}} \ nolimits _ {{4}} \ left) -P \ left (A \ mathop {{}} \ nolimits _ {{1}} A \ mathop {{}} \ nolimits _ {{2}} \ cap A \ mathop {{}} \ nolimits _ {{3}} A \ mathop {{}} \ NoLimits _ {{. 4}} \ right) \ right. \ right. \ right. \ right.} \)
\ ({= P \ mathop {{}} \ NoLimits ^ {{2}} + P \ mathop {{ }} \ nolimits ^ {{2 }} - P \ mathop {{}} \ nolimits ^ {{2}} P \ mathop {{}} \ nolimits ^ {{2}}} \)
Example 2: Suppose each probability of containing hepatitis virus in human serum, 0.4% probability of containing hepatitis virus seeking mixing 100 individual serum.
Set A = {contained in the serum hepatitis virus} then P (A) = 0.4%, then P {containing no hepatitis virus} = 99.6%
P (A) =. 1-P {containing no hepatitis virus}
\ ({= 1-P \ left (\ mathop {{\ cap}} \ limits _ {{i = 1}} ^ {{100}} \ left) \ overline {A} \ mathop {{}} \ nolimits _ {{i}} \ right. \ right.} \)
\ (1-0.996 = {\ mathop {} {} \ NoLimits ^ {}} 100 {} \)