Try the fastest way to create flashcards
Question

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

Verified
Step 1
1 of 3

Given:

σ=Standard deviation=600\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(All X>10000)=99%=0.99P(\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 Multiplication rule\textbf{Multiplication rule} for independent events: P(AB)=P(A and B)=P(A)×P(B)P(A\cap B)=P(A\text{ and }B)=P(A)\times P(B)

(P(X>10000))3=P(All X>10000)=99%=0.99(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)=0.9930.996655P(X>10000)=\sqrt[3]{0.99}\approx 0.996655

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

P(X10000)=1P(X>10000)=10.996655=0.003345P(X\leq 10000)=1-P(X>10000)=1-0.996655=0.003345

Create a free account to view solutions

Create a free account to view solutions

Recommended textbook solutions

Probability and Statistics for Engineers and Scientists 9th Edition by Keying E. Ye, Raymond H. Myers, Ronald E. Walpole, Sharon L. Myers

Probability and Statistics for Engineers and Scientists

9th EditionISBN: 9780321629111 (4 more)Keying E. Ye, Raymond H. Myers, Ronald E. Walpole, Sharon L. Myers
1,204 solutions
Applied Statistics and Probability for Engineers 3rd Edition by Douglas C. Montgomery, George C. Runger

Applied Statistics and Probability for Engineers

3rd EditionISBN: 9780471204541Douglas C. Montgomery, George C. Runger
1,384 solutions
Probability and Statistics for Engineers and Scientists 4th Edition by Anthony J. Hayter

Probability and Statistics for Engineers and Scientists

4th EditionISBN: 9781111827045 (2 more)Anthony J. Hayter
1,341 solutions
Applied Statistics and Probability for Engineers 6th Edition by Douglas C. Montgomery, George C. Runger

Applied Statistics and Probability for Engineers

6th EditionISBN: 9781118539712 (1 more)Douglas C. Montgomery, George C. Runger
2,027 solutions

More related questions

1/4

1/7