How Do You Calculate Battery Life Probability After 180 Hours?

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The discussion focuses on calculating the conditional probability of a battery lasting at least another 50 hours after already functioning for 180 hours. The battery life follows an exponential distribution with a probability density function of f(x) = (1/1000)e^(-x/1000) for x ≥ 0. To find the desired probability, one must apply the concept of conditional probability, specifically Pr(A|B), where A is the event of lasting at least 230 hours and B is the event of lasting at least 180 hours. The integration of the probability density function is required to determine the appropriate boundaries for the calculation. The key challenge lies in correctly setting the limits for the integration to reflect the conditions of the problem.
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Homework Statement


The life (in hours) of a particular brand of batteries is a random variable with probability density function given by f(x) = {1/1000e^(-x/1000)}, x ≥ 0, 0 elsewhere.

If after 180 hours of operation a battery is still working, what is the probability that it will last at least another 50 hours?


Homework Equations


Integration


The Attempt at a Solution


Integration of the equation, which becomes [-e^(-x/1000)]
Problem is, what boundaries is it supposed to have?
I have figured out from 0 --> 180, that is a given right? Because the question states 'if after 180 hours of operation', so it must have worked up to 180 hours.
but now what? I am sure it has got to do with finding the Probability (B) | Probability (A), which is equal to Pr (A|B) = Pr (A intersect B) / Pr (B)
 
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This is a conditional probability question: given that it has lasted 180 hours (that's the event on which to condition) you need to know the probability it will last another 50 hours.
 

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