91Ó°ÊÓ

In probability theory, the Complement Rule is a fundamental concept. It is especially useful when you want to find the probability that a certain event does not occur. The rule states that the probability of the complement of an event is equal to one minus the probability of the event itself. Mathematically, this is expressed as:

• \( P(A^c) = 1 - P(A) \)

This rule helps when calculating probabilities indirectly. For example, if you know the probability of the system having a type 1 defect is \(0.12\), you can determine the probability it does not have a type 1 defect by applying the Complement Rule. The calculation would be:

• \( P(A_1^c) = 1 - 0.12 = 0.88 \)

This means there is an 88% chance that the system will not have this defect. This approach simplifies many probability problems, especially when the event itself might be complicated to compute directly.

The concept of the Union of Events is related to finding the probability of either one or another event happening. In mathematical terms, the union of events \( A \) and \( B \) is written as \( A \cup B \), which includes all outcomes that are in \( A \), or \( B \), or in both. The Probability of Union of Events is calculated using the formula:

• \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)

This formula accounts for double-counting the scenarios in which both events occur. For instance, to calculate the probability that a system has either a type 1 or a type 2 defect, we use the given probabilities:

• \( P(A_1 \cup A_2) = P(A_1) + P(A_2) - P(A_1 \cap A_2) \)
• \( 0.13 = 0.12 + 0.07 - P(A_1 \cap A_2) \)
• \( P(A_1 \cap A_2) = 0.06 \)

The result illustrates how we eliminate overlaps to accurately calculate the union probability.

The Intersection of Events refers to the probability of two or more events occurring simultaneously. The intersection of events \( A \) and \( B \) is written as \( A \cap B \). It symbolizes all outcomes that occur in both events. This fundamental operation is key in determining the probability of combinations of events.

• For example, to find the probability that a system experiences both type 1 and type 2 defects while not having a type 3 defect, we start with:
• \( P((A_1 \cap A_2) \cap A_3^c) = P(A_1 \cap A_2) - P(A_1 \cap A_2 \cap A_3) \)
• \( P((A_1 \cap A_2) \cap A_3^c) = 0.06 - 0.01 = 0.05 \)

This indicates a 5% probability that both type 1 and type 2 defects occur without a defect of type 3. The intersection helps in narrowing down specific combinations of events, giving us a clear picture of coincidental occurrences.

The concept of "Probability of At Most Two Events" is crucial when considering a situation where a limited number of outcomes or occurrences is desired. This involves calculating the probability that at most two specific events happen simultaneously, meaning zero, one, or two events might occur, but not all three.

• To find this, you would subtract the probability of all three events occurring from the total probability:\( P(\text{at most two defects}) = 1 - P(A_1 \cap A_2 \cap A_3) \)
• With given values: \( P(\text{at most two defects}) = 1 - 0.01 = 0.99 \)

This calculation shows there's a 99% probability that the system does not have all three defects. Identifying such probabilities is beneficial when assessing risks and understanding the likelihood of multiple event combinations.