Question

practice calculating and applying conditional probabilit
what statements are correct? check all that apply.
the conditional probability formula is $p(x \mid y) = \frac{p(x \cap y)}{p(y)}$.
the conditional probabilities $p(d \mid n)$ and $p(n \mid d)$ are equal for any events d and n.
the notation $p(r \mid s)$ indicates the probability of event r, given that event s has already occurred.
conditional probability applies only to independent events.
conditional probabilities can be calculated using a venn diagram.

Explanation:

1. For the first statement: The conditional probability formula is indeed \( P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)} \) (when \( P(Y)

eq 0 \)), so this is correct.

1. For the second statement: \( P(D \mid N)=\frac{P(D\cap N)}{P(N)} \) and \( P(N \mid D)=\frac{P(D\cap N)}{P(D)} \). These are equal only if \( P(N) = P(D) \), not for any events \( D \) and \( N \), so this is incorrect.
2. For the third statement: The notation \( P(R \mid S) \) means the probability of \( R \) given that \( S \) has occurred, which is the definition of conditional probability notation, so this is correct.
3. For the fourth statement: Conditional probability applies to dependent events (and also can be used for independent events, but it's not restricted to only independent events). In fact, for independent events \( P(A\mid B)=P(A) \), but conditional probability is not limited to independent events, so this is incorrect.
4. For the fifth statement: A Venn diagram can help visualize the sample space and the intersections of events, which can be used to calculate conditional probabilities (by finding the ratio of the relevant regions), so this is correct.

Answer:

The correct statements are:

• The conditional probability formula is \( P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)} \).
• The notation \( P(R \mid S) \) indicates the probability of event \( R \), given that event \( S \) has already occurred.
• Conditional probabilities can be calculated using a Venn diagram.