Question

Twenty five heat lamps are connected in a greenhouse so, that when one lamp fails, another takes over immediately (only one lamp is turned on at any time.) The lamps operate independently, each with mean lifetime of 50 hours, and standard deviation of 4 hours. If the greenhouse is not checked for 1300 hours after the lamp system is turned on, what is the probability, that a lamp will be burning at the end of the 1300-hour period?

Answer 1

Answer:

P = 0.006

Step-by-step explanation:

Given

n = 25 Lamps

each with mean lifetime of 50 hours and standard deviation (SD) of 4 hours

Find probability that the lamp will be burning at end of 1300 hours period.

As we are not given that exact lamp, it means we have to find the probability where any of the lamp burning at the end of 1300 hours, So we have

Suppose i represents lamps

P (∑i from 1 to 25 ($X_i$ > 1300)) = 1300

= P($X^{'}$> $\frac{1300}{25}$)                  where $X^{'}$ represents mean time of a single lamp

= P (Z> $\frac{52-50}{\frac{1}{\sqrt{25}}}$)                Z is the standard normal distribution which can be found by using the formula

Z = Mean Time ($X^{'}$) - Life time of each Lamp (50 hours)/ (SD/$\sqrt{n}$)

Z = (52-50)/(4/$\sqrt{25}$) = 2.5

Now, P(Z>2.5) = 0.006 using the standard normal distribution table

Probability that a lamp will be burning at the end of 1300 hours period is 0.006