Question

How do you determine if the outcome of two events is independent?(1 point) Responses Show that P(A and B)=P(B)P(A) . Show that  upper P left parenthesis upper A and upper B right parenthesis equals Start Fraction upper P left parenthesis upper B right parenthesis over upper P left parenthesis upper A right parenthesis End Fraction . Show that P(A and B)=P(A)+P(B) . Show that  upper P left parenthesis upper A and upper B right parenthesis equals upper P left parenthesis upper A right parenthesis plus upper P left parenthesis upper B right parenthesis . Show that P(A and B)=P(A)⋅P(B) . Show that  upper P left parenthesis upper A and upper B right parenthesis equals upper P left parenthesis upper A right parenthesis times upper P left parenthesis upper B right parenthesis . Show that P(A and B)=P(A)P(B) . Show that  upper P left parenthesis upper A and upper B right parenthesis equals Start Fraction upper P left parenthesis upper A right parenthesis over upper P left parenthesis upper B right parenthesis End Fraction .

Answer 1

To determine if the outcome of two events \(A\) and \(B\) is independent, you would use the definition of independence in probability. Specifically, two events \(A\) and \(B\) are considered independent if and only if:

\[
P(A \text{ and } B) = P(A) \cdot P(B)
\]

This means that the probability of both events happening together (the joint probability \(P(A \text{ and } B)\)) equals the product of their individual probabilities.

So, among the provided responses, the correct statements that indicate the independence of events \(A\) and \(B\) are:

- Show that \(P(A \text{ and } B) = P(A) \cdot P(B)\).
- Show that \(P(A \text{ and } B) = P(B) \cdot P(A)\).

Both of these statements convey the same relationship and demonstrate that the events are independent.