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In probability theory, events are considered independent when the occurrence of one event does not influence the outcome of another. This is an essential concept when calculating probabilities for situations where multiple events occur simultaneously.
For instance, in the sports car example, defective brakes and a defective steering mechanism are independent events. This means that the likelihood of one defect happening does not affect the likelihood of the other defect occurring.
Understanding independence is crucial because it allows us to use the product rule for independent events. When events are independent, the probability of both occurring is simply the product of their individual probabilities.

The probability of a union of two events, often noted as \( P(A \cup B) \), refers to the likelihood of at least one of the events happening. For independent events, this is crucial when we want to know the probability that either one of the events or both occur.
In mathematical terms, the formula is:

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

This formula accounts for the fact that the probability of both events occurring (the intersection) is subtracted to avoid double-counting.
Applying this to the sports car scenario, where a lemon is defined as having at least one defect, makes the calculation clear. You add the probabilities of each defect occurring individually and then subtract the probability of both defects happening together.

The probability of intersection, denoted as \( P(A \cap B) \), represents the likelihood of both events occurring simultaneously.
When dealing with independent events, calculating the intersection is straightforward:

• Multiply the individual probabilities of each event.

For example, in our car problem, the probability that it is a hazard (both brakes and steering are defective) involves multiplying the probability of each defect: 0.15 for brakes and 0.05 for steering. Thus, the intersection probability ensures we see how likely both issues are to occur together. This type of calculation is simple yet a fundamental part of understanding joint probability in scenarios involving multiple risk factors.