Question

practice calculating and applying conditional probabilities. what statements are correct? check all that apply. the conditional probability formula is ( p(x | y)=\frac{p(x cap y)}{p(y)} ). 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. conditional probability applies only to independent events. conditional probabilities can be calculated using a venn diagram.

Explanation:

Step1: Analyze the first statement

The formula for conditional probability is \(P(X|Y)=\frac{P(X\cap Y)}{P(Y)}\) (when \(P(Y)> 0\)), so the first statement is correct.

Step2: Analyze the second statement

\(P(D|N)=\frac{P(D\cap N)}{P(N)}\) and \(P(N|D)=\frac{P(D\cap N)}{P(D)}\). In general, \(P(N)
eq P(D)\), so \(P(D|N)\) and \(P(N|D)\) are not equal for any events \(D\) and \(N\). The second statement is incorrect.

Step3: Analyze the third statement

By the definition of conditional - probability notation, \(P(R|S)\) means the probability of event \(R\) given that event \(S\) has already occurred. The third statement is correct.

Step4: Analyze the fourth statement

Conditional probability applies to both independent and dependent events. For independent events \(A\) and \(B\), \(P(A|B) = P(A)\), but it still follows the formula \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). The fourth statement is incorrect.

Step5: Analyze the fifth statement

In a Venn diagram, if we know the number of elements in the intersection of two sets and the number of elements in the given set, we can calculate the conditional probability (using the formula \(P(X|Y)=\frac{n(X\cap Y)}{n(Y)}\) which is analogous to the probability formula \(P(X|Y)=\frac{P(X\cap Y)}{P(Y)}\)). The fifth statement is correct.

Answer:

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