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To calculate probabilities in scenarios where events occur independently, you can multiply the individual probabilities of each event. Let's consider the exercise where a trading company uses eight computers. The probability of a single computer failing in one day is given as \(0.005\).

The goal in the exercise was to determine the probability that all eight computers fail on the same day. Each computer acts independently, which means the probability of all of them failing can be found by raising the individual failure probability to the power of the number of computers. Therefore, the formula looks like this:

• The probability for all eight computers failing: \[(0.005)^8 = 3.90625 \times 10^{-23}\]

This result allows us to understand the incredibly low likelihood that all computers will experience failure on the same day. This calculation is a direct application of the rule for independent probabilities.

Expected value, in simple terms, is the average expected outcome of a random event if it was repeated many times. In a geometric distribution, expected value helps us determine the average number of trials until an event—such as a computer failing—happens.

For a geometric distribution, which applies when events are independent and have equal probability each time, the expected number of trials (or days, in this context) until the first success (or failure) is calculated as:

• \(\frac{1}{p}\)

Here, \(p\) is the probability of failure for one computer in a day, which is \(0.005\). Therefore, the mean number of days until a specific computer fails is \(\frac{1}{0.005} = 200\) days.

In another scenario from the exercise where we need all eight computers to fail simultaneously, the probability changes to \(3.90625 \times 10^{-23}\) (as calculated earlier). Thus, the expected mean number of days until all computers fail on the same day is:

• \(\frac{1}{3.90625 \times 10^{-23}} \approx 2.56 \times 10^{22}\) days

This illustrates how rare simultaneous failures are, given the extremely high expected days calculated.

Understanding independent events is critical to accurately computing probabilities like we did in this exercise. An event is independent if the occurrence of any one event doesn't affect the probability of another event occurring. This property holds true for the computers in the trading company as each computer's operation is unaffected by the state of the others.

When dealing with independence:

• If Event A and Event B are independent, the probability of both occurring is the product of their probabilities: \(P(A \cap B) = P(A) \times P(B)\).

For the trading company's situation, because each computer failing is an independent event, we used this rule to calculate the probability of simultaneous failures, multipliyed the failure probabilities for all eight computers.

Recognizing and applying the principle of independent events is crucial in probability calculations and allows us to analyze complex scenarios effectively.