# A breakage of the electrical circuit occurs if either an element K_1 or both elements K_2 and K_3 break. The elements K_1, K_2, K_3 break independently of each other with probabilities respectively 0.3, 0.2, 0.1. Calculate the probability of circuit breakage.

Calculate the probability of circuit breakage.
A breakage of the electrical circuit occurs if either an element ${K}_{1}$ or both elements ${K}_{2}$ and ${K}_{3}$ break. The elements ${K}_{1},{K}_{2},{K}_{3}$ break independently of each other with probabilities respectively 0.3, 0.2, 0.1. Calculate the probability of circuit breakage.
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Step 1
Let Events ${A}_{1},{A}_{2},{A}_{3}$ denote the failure of ${K}_{1},{K}_{2},{K}_{3}$ resp.
Step 2
Probability that system fails
$P\left({A}_{1}\cap {\overline{A}}_{2}\cap {\overline{A}}_{3}\right)+P\left({A}_{1}\cap {A}_{2}\cap {A}_{3}\right)+P\left({\overline{A}}_{1}\cap {A}_{2}\cap {A}_{3}\right)$
$=P\left({A}_{1}\cap {\overline{A}}_{2}\cap {\overline{A}}_{3}\right)+P\left({A}_{2}\cap {A}_{3}\right)$
$=\left(0.3\right)\left(0.8\right)\left(0.9\right)+\left(0.2\right)\left(0.1\right)$