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?

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