Question
An electronic assembly consists of two subsystems, say, $A$ and $B.$ From 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 $P(A$ fails $B$ has failed$).$

Answer

Let's define events;
$E_{A }: A$ fails
$E_{B }: B$ fails
Given that:
Event failed by $A, P(E_A) = 0.2$
Event failed by both, $P\left(E_{A} \cap E_{B}\right) = 0.15$
And, event failed by B alone = $P\left(E_{B}\right)-P\left(E_{A} \cap E_{B}\right)$
$0.15 = P (E_B) - 0.15$
$\therefore P (E_B) = 0.30$
Therefore, $P\left(E_{A} | E_{B}\right)=\frac{P\left(E_{A} \cap E_{B}\right)}{P\left(E_{B}\right)}$
$= \frac{0.15}{0.3}$
$= 0.5$
Which is the required solution.

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