Verified Answer :

  • Events \(A\) and \(B\) are independent if the occurrence of one does not affect the other.
  • The fundamental property of independence is \(P(A \cap B) = P(A)P(B)\).
  • Using the definition of conditional probability, \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\), we substitute the property to find \(P(A \mid B) = \frac{P(A)P(B)}{P(B)} = P(A)\).
  • Therefore, the correct statement is \(\boxed{P(A \mid B) = P(A)}\).

Explanation

  1. Defining independent events

In probability theory, two events, \(A\) and \(B\), are defined as independent if the occurrence of one event does not influence the likelihood of the other event occurring. Formally, this means that the conditional probability of \(A\) given that \(B\) has occurred is simply the probability of \(A\) itself, assuming \(P(B) > 0\). This is expressed mathematically as \(P(A \mid B) = P(A)\). Similarly, it implies \(P(B \mid A) = P(B)\).

  1. Deriving the relationship from the definition of independence

Let's analyze the definition of conditional probability:
\[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\]
If \(A\) and \(B\) are independent, the joint probability \(P(A \cap B)\) is equal to the product of their individual probabilities:
\[P(A \cap B) = P(A)P(B)\]
Substituting this into the conditional probability formula, we get:
\[P(A \mid B) = \frac{P(A)P(B)}{P(B)} = P(A)\]
This confirms that the statement \(P(A \mid B) = P(A)\) is a direct consequence of the definition of independence.

  1. Evaluating the choices

Now we evaluate the provided options:

  1. \(P(A \mid B) = P(B)\): This is generally false, as it implies a specific relationship between \(P(A)\) and \(P(B)\) that does not hold for independent events in general.
  2. \(P(A \mid B) = P(A)\): This is the correct definition/consequence of independence.
  3. \(P(A) = P(B)\): This is not required for independence; events can be independent and have different probabilities.
  4. \(P(A \mid B) = P(B \mid A)\): This is generally false, as it would imply \(P(A) = P(B)\), which is not a requirement for independence.

Thus, the correct condition is \(P(A \mid B) = P(A)\).

Examples

Understanding independent events is crucial in fields like quality control or risk assessment. For example, if you are testing two different components on a machine and you know their failure rates are independent, the probability that the first one fails, given that the second one has failed, remains exactly the same as the probability that the first one fails on its own. This helps engineers calculate the overall reliability of a system by simply multiplying individual probabilities together.

Related Questions