A trading company has eight computers that it uses to trade...

A trading company has eight computers that it uses to trade on the New York Stock Exchange (NYSE). The probability of a computer failing in a day is $0.005,$ and the computers fail independently. Computers are repaired in the evening and each day is an independent trial. (a) What is the probability that all eight computers fail in a day? (b) What is the mean number of days until a specific computer fails? (c) What is the mean number of days until all eight computers fail in the same day?

A trading company has eight computers that it uses to trade on the New York Stock Exchange (NYSE). The probability of a computer failing in a day is $0.005,$ and the computers fail independently. Computers are repaired in the evening and each day is an independent trial.
(a) What is the probability that all eight computers fail in a day?
(b) What is the mean number of days until a specific computer fails?
(c) What is the mean number of days until all eight computers fail in the same day?

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Key Concepts

Independent Events

This concept refers to situations where the occurrence of one event does not affect the occurrence of another. In probability, when events are independent, the probability of all events occurring together is the product of their individual probabilities.

Multiplication Rule for Independent Events

This rule states that if multiple events occur independently, the probability that all events occur is the product of their separate probabilities. This is essential for calculating the joint probability of several independent events happening together.

Geometric Distribution

The geometric distribution models the number of trials needed for the first success in a sequence of independent Bernoulli trials, where each trial has the same probability of success. It is characterized by its mean, which is the reciprocal of the success probability.

Expected Value

The expected value, or mean, in probability theory is a measure of the central tendency of a random variable. In the context of the geometric distribution, it represents the average number of trials required for an event to occur for the first time, calculated as the inverse of the success probability.

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