Upgrading a certain software package requires installation of 68 new files. Files are installed consecutively. The installation

Question

2 months ago

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Upgrading a certain software package requires installation of 68 new files. Files are installed consecutively. The installation

time is random, but on the average, it takes 15 sec to install one file, with a variance of 11 sec2. (a) What is the probability that the whole package is upgraded in less than 12 minutes?

Mathematics

1 answer:

lawyer [12.5K] · Answer 1 · 2026-03-09T10:25:31+03:00

Answer:

The chance of completing the entire package installation in under 12 minutes is 0.1271.

Step-by-step explanation:

We define X as a normal distribution representing the time taken in seconds to install the software. According to the Central Limit Theorem, X is approximately normal, where the mean is 15 and variance is 15, giving a standard deviation of √15 = 3.873.

To find the probability of the total installation lasting less than 12 minutes, which equals 720 seconds, each installation should average under 720/68 = 10.5882 seconds. Thus, we seek the probability that X is less than 10.5882. To do this, we will apply W, the standard deviation value of X, calculated via the formula provided.

Utilizing $\phi$ , we reference the cumulative distribution function of the standard normal variable W, with values found in the attached file.

$P(X < 10.5882) = P(\frac{X-15}{3.873} < \frac{10.5882-15}{3.873}) = P(W < -1,14)$

Given the symmetry of the standard normal distribution density function, we ascertain $\phi(-1.14) = 1-\phi(1.14) = 1-0.8729 = 0.1271$ .

Consequently, the probability that the installation process for the entire package is completed within 12 minutes is 0.1271.

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