The lifetime of a light bulb is described as an exponential random variable with a mean of 1000 hours. This implies that the rate parameter (λ) of the exponential distribution can be calculated as λ = 1/mean = 1/1000. Thus, λ = 0.001.
To find the probability that the light bulb lasts more than 900 hours, we can use the cumulative distribution function (CDF) of an exponential distribution. The CDF of an exponential random variable X is given by:
P(X “> x) = e^{-λx}
We start by calculating P(X > 900):
P(X > 900) = 1 – P(X ≤ 900) = e^{-λ*900} = e^{-0.001*900}
Calculating the exponent:
-0.001 * 900 = -0.9
So, we have:
P(X > 900) = e^{-0.9}
Using a calculator or an exponential function table, we find:
e^{-0.9} ≈ 0.40657
Now, this value represents the probability of the light bulb lasting more than 900 hours. Since we are given that the light bulb is guaranteed to last at least 900 hours, we need to find the conditional probability that it lasts more than 1000 hours given that it has already lasted 900 hours. This is given by:
P(X > 1000 | X > 900) = P(X > 1000) / P(X > 900)
We already know:
P(X > 900) ≈ 0.40657
Now, we calculate P(X > 1000):
P(X > 1000) = e^{-0.001*1000} = e^{-1} ≈ 0.36788
Finally, we can compute the conditional probability:
P(X > 1000 | X > 900) = P(X > 1000) / P(X > 900) ≈ 0.36788 / 0.40657 ≈ 0.90516
Thus, the probability that the light bulb will last longer than its mean lifetime of 1000 hours, given that it has already lasted at least 900 hours, is approximately 0.905, or 90.5%.