Question

Suppose the probability that a particular computer chip fails after \(a\) hours of operation is \(0.00005 \int_{a}^{\infty} e^{-0.00005 t} d t\) a. Find the probability that the computer chip fails after \(15,000 \mathrm{hr}\) of operation. b. Of the chips that are still operating after \(15,000 \mathrm{hr}\), what fraction of these will operate for at least another \(15,000 \mathrm{hr} ?\) c. Evaluate \(0.00005 \int_{0}^{\infty} e^{-0.00005 t} d t\) and interpret its meaning.

Short answer

Answer: The probability that the computer chip fails after 15,000 hours of operation is approximately 0.5276. Of the chips that are still operating after 15,000 hours, approximately 1.64 times as many will operate for at least another 15,000 hours.

Step-by-step solution

Plug in the values into the integral

To find the probability of the computer chip failing after 15,000 hours of operation, replace "a" with 15,000 in the integral and evaluate: \(0.00005 \int_{15000}^{\infty} e^{-0.00005 t} d t\)

Evaluate the indefinite integral

Evaluate the integral of \(e^{-0.00005 t}\) with respect to t: \(\int e^{-0.00005 t} dt = -\frac{1}{0.00005} e^{-0.00005 t} + C\)

Evaluate the definite integral

Insert the limits of integration (15000 and \(\infty\)) into the indefinite integral and calculate the resulting value: \(-\frac{1}{0.00005} \left[ e^{-0.00005 (\infty)} - e^{-0.00005 (15000)} \right] = -\frac{1}{0.00005} \left[ 0 - e^{-0.75} \right]\)

Calculate the probability

Multiply the result of the definite integral by 0.00005 to find the probability of failure after 15,000 hours: \(0.00005 \times -\frac{1}{0.00005} \left[ 0 - e^{-0.75} \right] = 1 - e^{-0.75} \approx 0.5276\) The probability that the computer chip fails after 15,000 hours of operation is approximately 0.5276. b. Fraction of chips operating for at least another 15,000 hours

Calculate the probabilities

Find the probability of the chip still operating after 15,000 hours, which is the complement of the probability found in part (a). \(P(operating\ after\ 15000\ hrs) = 1 - 0.5276 = 0.4724\) Find the probability of the chip still operating after 30,000 hours by evaluating the integral with "a" as 30,000: \(P(operating\ after\ 30000\ hrs) = 1 - e^{-0.00005\times30000} = 1 - e^{-1.5} \approx 0.7769\)

Find the fraction

Divide the probability of operating after 30,000 hours by the probability of operating after 15,000 hours to get the desired fraction: \(\frac{P(operating\ after\ 30000\ hrs)}{P(operating\ after\ 15000\ hrs)}=\frac{0.7769}{0.4724}\approx1.64\) Of the chips that are still operating after 15,000 hours, approximately 1.64 times as many will operate for at least another 15,000 hours. c. Evaluation and meaning of the integral

Change the limits to 0 and infinity

Replace the limits of integration in the integral with 0 and infinity: \(0.00005 \int_{0}^{\infty} e^{-0.00005 t} d t\)

Evaluate the definite integral

Insert the limits of integration (0 and \(\infty\)) into the indefinite integral -\frac{1}{0.00005} e^{-0.00005 t} and calculate the resulting value: \(-\frac{1}{0.00005} \left[ e^{-0.00005 (\infty)} - e^{-0.00005 (0)} \right] = -\frac{1}{0.00005} \left[ 0 - 1 \right]\)

Calculate the integral value

Multiply the result of the definite integral by 0.00005: \(0.00005 \times -\frac{1}{0.00005} \left[ 0 - 1 \right] = 1\) The value of the integral is 1, which indicates that the total probability of the computer chip failing over its entire operational time is 100%. This result makes sense, as the sum of all possible probabilities of failure over time must equal 1.

Key concepts

Indefinite Integral

Understanding the indefinite integral is crucial for solving many problems in calculus, including the probability of a computer chip failing over time. An indefinite integral is a general form of integration without specified limits, represented as \[ \int f(x) dx = F(x) + C \], where \( F(x) \) is the antiderivative of \( f(x) \), and \( C \) is the constant of integration. The process of finding the indefinite integral is known as anti-differentiation. It helps us to determine a function when its rate of change is known.

In the context of the problem, the indefinite integral is the first step towards finding the cumulative probability distribution of the computer chip failure, which describes how the probability accumulates over time. By solving the indefinite integral \[ \int e^{-0.00005 t} dt \], we are setting the stage to compute the definite integral, which will give us the exact probabilities needed for different time intervals.

Exponential Decay

The concept of exponential decay is a fundamental mathematical model used to describe the process by which a quantity decreases rapidly at a rate proportional to its current value. In formulas, this is often represented as \[ f(t) = e^{-kt} \], where \( e \) is the base of the natural logarithm, \( t \) is time, and \( k \) is a positive constant representing the rate of decay.

In the exercise provided, the probability of computer chip failure is modeled using exponential decay, where the decay constant is inversely related to the chip's reliability over time. As time increases, the exponential term decreases, indicating a lower probability of failure at that moment, consistent with the expected behavior of reliable electronic components. This model is crucial for reliability engineering and helps in making predictions about a product's lifespan and performance.

Definite Integral Evaluation

The evaluation of a definite integral involves calculating the area under a curve between two specified limits, often representing a total quantity across an interval. Represented mathematically as \[ \int_{a}^{b} f(x) dx = F(b) - F(a) \], it takes the antiderivative \( F(x) \) found from the indefinite integral and uses the Fundamental Theorem of Calculus to obtain a definite (numeric) value.

In our exercise, evaluating the definite integral from 15,000 hours to infinity provides the probability that a computer chip will fail after 15,000 hours. By taking the limits of the antiderivative and computing the difference, we translate the general formula of the probability distribution into a specific value. This serves as an essential technique in probability and statistics for determining the likelihood of events over a finite period. Understanding how to evaluate definite integrals is crucial for any student looking to work with probabilities in practical, real-world situations.