Questions: A lamp has two light bulbs with an average lifespan of 1000 hours. Assuming we can model the probability of failure of these bulbs by an exponential density function with mean μ=1000, find the probability that both of the lamp's bulbs fail within 500 hours. Express the answer as a decimal rounded to four decimal places.

A lamp has two light bulbs with an average lifespan of 1000 hours. Assuming we can model the probability of failure of these bulbs by an exponential density function with mean μ=1000, find the probability that both of the lamp's bulbs fail within 500 hours. Express the answer as a decimal rounded to four decimal places.
Transcript text: A lamp has two light bulbs with an average lifespan of 1000 hours. Assuming we can model the probability of failure of these bulbs by an exponential density function with mean $\mu=1000$, find the probability that both of the the lamp's bulbs fail within 500 hours. Express the answer as a decimal rounded to four decimal places.
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Solution

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Solution Steps

Step 1: Define the Exponential Distribution

The lifespan of the lamp's bulbs is modeled by an exponential distribution with a mean \( \mu = 1000 \) hours. The rate parameter \( \lambda \) is given by:

\[ \lambda = \frac{1}{\mu} = \frac{1}{1000} = 0.001 \]

Step 2: Calculate the Probability for One Bulb

To find the probability that one bulb fails within 500 hours, we use the cumulative distribution function (CDF) of the exponential distribution:

\[ F(x; \lambda) = 1 - e^{-\lambda x} \]

Calculating \( P(0 \leq X \leq 500) \):

\[ P(0 \leq X \leq 500) = F(500) - F(0) = \left(1 - e^{-0.001 \cdot 500}\right) - \left(1 - e^{-0.001 \cdot 0}\right) \]

This simplifies to:

\[ P(0 \leq X \leq 500) = 1 - e^{-0.5} - 0 = 1 - e^{-0.5} \approx 0.3935 \]

Thus, the probability that one bulb fails within 500 hours is:

\[ P(\text{one bulb fails within 500 hours}) \approx 0.3935 \]

Step 3: Calculate the Probability for Both Bulbs

Since the failures of the bulbs are independent events, the probability that both bulbs fail within 500 hours is the product of the individual probabilities:

\[ P(\text{both bulbs fail within 500 hours}) = P(\text{one bulb fails})^2 = (0.3935)^2 \approx 0.1548 \]

Final Answer

The probabilities are as follows:

  • Probability that one bulb fails within 500 hours: \( \approx 0.3935 \)
  • Probability that both bulbs fail within 500 hours: \( \approx 0.1548 \)

Thus, the final answer is:

\[ \boxed{0.1548} \]

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