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# A product contains three lasers, and the product fails if any of the lasers fails. Assume the lasers fail independently. What should the mean life equal in order for 99% of the products to exceed 10,000 hours before failure?

Solution

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Given:

\begin{align*} \sigma&=\text{Standard deviation}=600 \end{align*}

99% of the products exceed a lifetime of 10000, while a product fails when at least one of the lasers fail.

$P(\text{All }X>10000)=99\%=0.99$

Assuming that the lasers fail independently, it is safe to assume that they are independent and thus it is appropriate to use the $\textbf{Multiplication rule}$ for independent events: $P(A\cap B)=P(A\text{ and }B)=P(A)\times P(B)$

$(P(X>10000))^3=P(\text{All }X>10000)=99\%=0.99$

Let us take the cubic root of each side of the previous equation:

$P(X>10000)=\sqrt[3]{0.99}\approx 0.996655$

Use the $\textbf{Complement rule}$: $P(A^c)=P(\text{ not }A)=1-P(A)$

$P(X\leq 10000)=1-P(X>10000)=1-0.996655=0.003345$

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