Question

An electronic assembly consists of two subsystems, say A and B. From the previous testing procedures, the following probabilities are assumed to be known:
P( A fails) = 0.2
P(B fails alone) = 0.15
P(A and B fail) = 0.15
Evaluate the following probabilities:
(i) P( A fails $left enclose straight B space has space failed end enclose$ ) (ii) P(A fails alone)

Solution

Consider the following events:
E : A fails
F : B fails.
Now P(A fails) = 0.2 $rightwards double arrow$ P(E) = 0.2
and P(A and B fails) = 0.15 $rightwards double arrow$ $straight P left parenthesis straight E intersection straight F right parenthesis space equals space 0.15$
Also P(B fails alone) = 0.15
$rightwards double arrow space space space straight P left parenthesis straight E with rightwards arrow on top space intersection space straight F right parenthesis space equals space 0.15 rightwards double arrow space space space straight P left parenthesis straight F right parenthesis space minus space straight P left parenthesis straight E space intersection space straight F right parenthesis space equals space 0.15 rightwards double arrow space space space space straight P left parenthesis straight F right parenthesis space equals space straight P left parenthesis straight E intersection straight F right parenthesis space plus space 0.15 rightwards double arrow space space space straight P left parenthesis straight F right parenthesis space equals space 0.15 plus 0.15 space equals space 0.30$
(i) P(A fails| B has failed) = P(E | F) = $fraction numerator straight P left parenthesis straight E intersection straight F right parenthesis over denominator straight P left parenthesis straight F right parenthesis end fraction space equals space fraction numerator 0.15 over denominator 0.30 end fraction space equals space 15 over 30 space equals space 1 half$
(ii) P(A fails alone) = $straight P left parenthesis straight E intersection straight F with rightwards arrow on top right parenthesis space equals space straight P left parenthesis straight E right parenthesis space minus space straight P left parenthesis straight E intersection straight F right parenthesis$
= 0.2 - 0.15 = 0.05

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