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 fails
E: B fails
Given that:
Event failed by A, P(EA) = 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 (EB) - 0.15
$\therefore$ P (EB) = 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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