A congested computer network has a 1 % chance of losing a...

A congested computer network has a $1 \%$ chance of losing a data packet, and packet losses are independent events. An e-mail message requires 100 packets. (a) What distribution of data packets must be re-sent? Include the parameter values. (b) What is the probability that at least one packet must be re-sent? (c) What is the probability that two or more packets must be re-sent? (d) What are the mean and standard deviation of the number of packets that must be re-sent? (e) If there are 10 messages and each contains 100 packets, what is the probability that at least one message requires that two or more packets be re-sent?

A congested computer network has a $1 \%$ chance of losing a data packet, and packet losses are independent events. An e-mail message requires 100 packets.
(a) What distribution of data packets must be re-sent? Include the parameter values.
(b) What is the probability that at least one packet must be re-sent?
(c) What is the probability that two or more packets must be re-sent?
(d) What are the mean and standard deviation of the number of packets that must be re-sent?
(e) If there are 10 messages and each contains 100 packets, what is the probability that at least one message requires that two or more packets be re-sent?

Key Concepts

Independent Events

In the study of probability, independent events refer to outcomes that do not affect each other. In the network example, each packet has an independent chance of being lost, meaning the outcome of one packet transmission has no impact on the outcomes of others.

Mean and Standard Deviation of the Binomial Distribution

For a binomial distribution, the mean is calculated as np, and the variance is np(1-p); the standard deviation is the square root of the variance. These metrics provide a measure of the central tendency and spread of the number of packet losses.

Complement Rule in Probability

The complement rule states that the probability of an event occurring is one minus the probability of the event not occurring. This rule is useful to determine the probability of at least one occurrence in a series of independent events, such as finding the chance that at least one packet is lost.

Bernoulli Trial

A Bernoulli trial is a random experiment with exactly two possible outcomes: success or failure. Here, each packet transmission is a Bernoulli trial, where a loss (or need to re-send) is considered a 'success' in counting errors, occurring with a fixed probability.

Binomial Distribution

A binomial distribution models the number of successes (or failures) in a fixed number of independent trials, where each trial has two outcomes and a constant probability of success. In this context, each data packet sent represents an independent trial with a fixed probability of being lost.

Parameters of the Binomial Distribution

The parameters n and p in a binomial distribution represent the number of independent trials (n) and the probability of success (p) on each trial, respectively. These parameters are used to calculate probabilities, the mean value, and the variance of the distribution.

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