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In probability theory, independent events are two or more events that do not affect each other's outcomes. When dealing with independent events, the occurrence of one event has no influence on the occurrence of another. For example, in the case of the transistors, each component's likelihood of being defective is unaffected by the state of the other components.

This property allows us to perform probability calculations by simply multiplying the probabilities of individual events.

• For an event A having a probability of occurrence, denoted as \(P(A)\).
• And event B being independent with probability \(P(B)\).

The probability of both events A and B occurring is calculated as \(P(A \cap B) = P(A) \times P(B)\).

In this exercise, each component is considered independently defective with a probability of 0.1. Thus, the probability of it being nondefective is 0.9. The fact that the components' states do not influence one another allows us to calculate the probability of all four components being nondefective by multiplying the probability of one being nondefective raised to the fourth power.

Complementary probability is a fundamental concept that helps us understand the relationship between an event and its complement. The complement of an event A is the event that A does not occur. The probability of the complement of event A is calculated as 1 minus the probability of event A occurring.

In mathematical terms, if \(P(A)\) is the probability of event A, then \(P(A^c) = 1 - P(A)\), where \(A^c\) denotes the complement of A.

In the context of this exercise, we're dealing with components that have a probability of being defective. Given that each component is independently defective with a probability of 0.1, the complementary probability of being nondefective is 0.9. This is helpful because we're interested in the scenario in which all four randomly inspected components are nondefective, or, in other words, in the complement of any one being defective.

Probability calculation involves determining the likelihood of a particular event or series of events. To calculate probabilities effectively, understanding the nature of the events, such as whether they are independent or dependent, is crucial.

In this instance of independent events, the steps to determine the overall probability are straightforward through multiplication.

• First, establish the probability of a single event happening, here being the event that a transistor is nondefective, which we found to be 0.9.
• Then apply the principle of independent events by multiplying this probability for each event in the series.

For the case of the 4 nondefective transistors, we computed \((0.9)^4 = 0.6561\), indicating a 65.61% probability of acceptance.

The rejection probability, being the complement, is found by subtracting the acceptance probability from 1, resulting in approximately 34.39%. This conversion of probability into percentage forms gives practical insight into decision-making scenarios like quality control, as showcased by the transistor purchasing policy.