Question

An undergraduate business student has purchased a laptop computer for use during exams. This laptop is perfectly reliable except for two parts: its microchip, which has a failure rate of one in every twenty hours of operation; and its battery, which has a failure rate of one in every ten hours of operation. Also, on average the battery will wear out in five hours, with a standard deviation of 30 minutes. Assuming that a new battery has just been installed, what is the probability that the laptop will perform reliably during a one-hour exam?

Answer 1

The overall probability that the laptop performs reliably during a one-hour exam is found by multiplying the independent probabilities of no failure for both the microchip and the battery, resulting in an 85.5% chance. The longevity of the battery also contributes positively to this likelihood due to its average life span of five hours with a small standard deviation.

Step-by-step explanation:

The student's question involves calculating the probability that a laptop will perform reliably during a one-hour exam, taking into account the failure rates of its microchip and battery. Specifically, the microchip has a failure rate of one in every twenty hours, and the battery has a failure rate of one in every ten hours. Additionally, the battery life expectancy is normally distributed with a mean of five hours and a standard deviation of 30 minutes (0.5 hours).

To find the overall probability of the laptop performing reliably during a one-hour exam, we must consider the reliability of both the battery and the microchip separately and then multiply those probabilities together since both must work for the laptop to perform reliably.

The probability of the microchip not failing in one hour is 19/20 or 0.95. The probability of the battery not failing in one hour is 9/10 or 0.9. Assuming these events are independent, the combined probability of neither component failing is 0.95 * 0.9 = 0.855.

Since the battery has a normal distribution for its life expectancy, we must also check the probability that the battery lasts more than one hour based on its mean and standard deviation. This can be calculated using a z-score formula or a standard normal distribution table. However, since the average life is five hours, which is much longer than the one-hour exam, and the standard deviation is relatively small, it is likely that the probability is very high that the newly installed battery will last for the duration of the exam. Therefore, the laptop is fairly reliable for at least the duration of the one-hour exam based on the given failure rates and distribution of battery life.

Answer 2

probability of a reliable performance = 19/20

Step-by-step explanation:

If a new battery has just been installed, then the chances of failure are only contributed by the faulty microchip, which has a failure rate of 1 in 20 hours.

Therefore in 1 hour, the probability that the laptop will perform reliably is calculated as follows:

Chances of failure = 1 in 20

20 hours = 1 chance of failure

∴ 1 hour = 1/20 chance of failure

1/20 = 0.05 chance of failure

Now, let us express 0.05 as fraction:

0.05 = 5/100

Therefore in 1 hour, there is a 5 in 100 chances of failure.

But we are asked to find the probability that the laptop will perform reliably. This is simply done by finding the difference between the total chances of occurrence (100) and the chances of failure (5)

∴ Probability of reliable performance in one hour = (100 - 5) / 100

= 95/100

95/100 = 19/20