Question

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

Explanation:

Step1: Analyze the first statement

The formula for conditional probability is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\) and \(P(B|A)=\frac{P(A\cap B)}{P(A)}\). These are equal only if \(P(A) = P(B)\), not for any events. So the first statement is wrong.

Step2: Analyze the second statement

Conditional probability applies to dependent and independent events. For independent events \(P(A|B)=P(A)\), but the concept of conditional probability is not restricted to independent events. So the second statement is wrong.

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. This is correct.

Step4: Analyze the fourth statement

The formula \(P(X|Y)=\frac{P(X\cap Y)}{P(Y)}\) (where \(P(Y)> 0\)) is the standard formula for conditional probability. This is correct.

Step5: Analyze the fifth statement

We can use a Venn diagram. The number of elements in \(X\cap Y\) divided by the number of elements in \(Y\) (for equally - likely outcomes) or the area of \(X\cap Y\) divided by the area of \(Y\) (in a probability - area model) can calculate conditional probability. This is correct.

Answer:

The third, fourth, and fifth statements are correct.