Final answer:
The lifetime of an electric bulb with an exponential distribution and mean of 180 days requires using the cumulative distribution function to find the probability of burning out within a certain time and the memoryless property for conditional probabilities.
Step-by-step explanation:
The lifetime of an electric bulb follows an exponential distribution with a mean life of 180 days. The probability density function of an exponential distribution with mean μ is f(x) = (1/μ) * e^{-x/μ} for x ≥ 0.
1. To find the probability that the bulb burns out within 160 days, we can use the cumulative distribution function (CDF), which for an exponential distribution is F(x) = 1 - e^{-x/μ}. Plugging in the mean μ = 180 and x = 160, we get F(160) = 1 - e^{-160/180}. This computation gives us the probability of the bulb burning out within 160 days.
2. Given that the bulb is still working after 90 days, we'll use the memoryless property of exponential distributions. The probability that the bulb lasts more than an additional 60 days, given that it has already lasted 90 days, is the same as the initial probability of it lasting 60 days. This is computed as P(X > 60) = e^{-60/μ}, substituting μ = 180 to find the probability.