Question

calculating and applying conditional probabilities
what statements are correct? choose three correct answers.
the conditional probability formula is ( p(x|y)=\frac{p(xcap y)}{p(y)} ).
conditional probability applies only to independent events.
conditional probabilities can be calculated using a venn diagram.
the conditional probabilities ( p(d|n) ) and ( p(n|d) ) are equal for any events d and n.
the notation ( p(r|s) ) indicates the probability of event r given that event s has already occurred.

Explanation:

• The formula \(P(X|Y)=\frac{P(X\cap Y)}{P(Y)}\) is the standard conditional probability formula.
• Conditional probability can be calculated using a Venn diagram by finding the ratio of the intersection region to the given event region.
• The notation \(P(R|S)\) indeed means the probability of event \(R\) given that event \(S\) has occurred.
• Conditional probability applies to both independent and dependent events. For independent events \(P(X|Y) = P(X)\) (since \(P(X\cap Y)=P(X)P(Y)\)), but it's still calculated using the conditional - probability formula.
• \(P(D|N)=\frac{P(D\cap N)}{P(N)}\) and \(P(N|D)=\frac{P(D\cap N)}{P(D)}\). These are equal only when \(P(D) = P(N)\), not for any events \(D\) and \(N\).

Answer:

The first box (The conditional probability formula is \(P(X|Y)=\frac{P(X\cap Y)}{P(Y)}\)), the third box (Conditional probabilities can be calculated using a Venn diagram), and the fifth box (The notation \(P(R|S)\) indicates the probability of event \(R\) given that event \(S\) has already occurred) are correct.