Question

Reconsider the credit card scenario of, and show that A and B are dependent first by using the definition of independence and then by verifying that the multiplication property does not hold.

Short answer

\({\rm{A}}\) and \({\rm{B}}\) are dependent.

Step-by-step solution

Definition

Independence is a state in which a person, a nation, a country, or a state's people and population, or a portion of them, have self-government and, in most cases, sovereignty over their area. The position of a dependent territory is the polar opposite of independence.

Proofing A and B are dependent

If for two events \({\rm{A}}\) and \({\rm{B}}\) stands

\(P(A\mid B) = P(A)\)

we say that they are independent. They are dependent otherwise.

From the exercise \({\rm{47}}\) we have that

\(P(A\mid B)\mathop = \limits^{(1)} \frac{{P(A \cap B)}}{{P(B)}}\mathop = \limits^{(1)} \frac{{0.3}}{{0.4}} = 0.75\)

(1): here we used the definition of conditional probability given below,

(1): we are given the probabilities in the exercise \({\rm{47}}{\rm{.}}\)

Conditional probability of A given that the event B has occurred, for which\({\rm{P(B) > 0}}\), is

\(P(A\mid B) = \frac{{P(A \cap B)}}{{P(B)}}\)for any two events \({\rm{A}}\)and \({\rm{B}}{\rm{.}}\)

Determining for events

We also have that

\({\rm{P(A) = 0}}{\rm{.6}}\)

Obviously

\(0.75 = P(A\mid B) \ne P(A) = 0.6,\)

which means that \({\rm{\backslash }}\) text events \({\rm{A}}\) and \({\rm{B}}\) are dependent \({\rm{\} \} }}\).

To verify it using the multiplication property:

Proposition: Two events \({\rm{A}}\) and \({\rm{B}}\)are independent if and only if

\(P(A \cap B) = P(A) \cdot P(B)\)

We have

\(\begin{array}{c}P(A \cap B) &=& 0.3\\P(A) \cdot P(B) &=& 0.6 \cdot 0.4\\ &=& 0.24\end{array}\)

From which the following is true

\(P(A \cap B) \ne P(A) \cdot P(B)\)

Therefore, using the proposition given above events \({\rm{A}}\) and \({\rm{B}}\) are dependent.