Question

if events a and b are independent, what must be true?

\\(p(a|b) = p(b)\\)
\\(p(a|b) = p(a)\\)
\\(p(a) = p(b)\\)
\\(p(a|b) = p(b|a)\\)

Explanation:

Define independent events

Using the Independent Events knowledge point

By definition, two events \(A\) and \(B\) are independent if the occurrence of one event does not affect the probability of the occurrence of the other. Mathematically, this is expressed as:

$$ P(A \cap B) = P(A) \cdot P(B) $$

Apply conditional probability formula

Using the Conditional Probability knowledge point

The conditional probability of event \(A\) given that event \(B\) has occurred is defined as:

$$ P(A|B) = \frac{P(A \cap B)}{P(B)} $$

Substitute independence into the conditional probability formula

Using the Independent Events and Conditional Probability knowledge points

Substituting \(P(A \cap B) = P(A) \cdot P(B)\) into the conditional probability formula:

$$ P(A|B) = \frac{P(A) \cdot P(B)}{P(B)} = P(A) $$

This confirms that the probability of \(A\) occurring given that \(B\) has occurred is simply the probability of \(A\).

Evaluate the given options

Using the Independent Events knowledge point

• \(P(A|B) = P(B)\): Incorrect, as the conditional probability of \(A\) given \(B\) depends on the probability of \(A\), not \(B\).
• \(P(A|B) = P(A)\): Correct, as derived from the definition of independence.
• \(P(A) = P(B)\): Incorrect, independent events do not need to have equal probabilities.
• \(P(A|B) = P(B|A)\): Incorrect, as \(P(A|B) = P(A)\) and \(P(B|A) = P(B)\), which are not necessarily equal.

Answer:

• (A) \(P(A|B) = P(B)\)
• (B) \(P(A|B) = P(A)\) (Correct answer)
• (C) \(P(A) = P(B)\)
• (D) \(P(A|B) = P(B|A)\)