practice calculating and applying conditional probabilities. what statements are correct? check all that…

practice calculating and applying conditional probabilities. what statements are correct? check all that apply. the conditional probability formula is $p(x|y)=\frac{p(xcap y)}{p(y)}$. 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.

practice calculating and applying conditional probabilities. what statements are correct? check all that apply. the conditional probability formula is $p(x|y)=\frac{p(xcap y)}{p(y)}$. 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.

Answer

Explanation:

Step1: Recall conditional - probability formula

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

Step2: Analyze $P(D|N)$ and $P(N|D)$

$P(D|N)=\frac{P(D\cap N)}{P(N)}$ and $P(N|D)=\frac{P(N\cap D)}{P(D)}$. They are not equal in general, so the second statement is incorrect.

Step3: Understand conditional - probability notation

The notation $P(R|S)$ indeed indicates the probability of event $R$ given that event $S$ has already occurred, so the third statement is correct.

Step4: Know the scope of conditional - probability

Conditional probability applies to both independent and dependent events. For independent events $A$ and $B$, $P(A|B) = P(A)$ and $P(B|A)=P(B)$, but it still makes sense to calculate conditional probabilities for them. So the fourth statement is incorrect.

Step5: Consider Venn - diagram usage

We can calculate conditional probabilities using a Venn diagram by finding the appropriate intersections and individual event probabilities. So the fifth statement is correct.

Answer:

The correct statements are:

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