Question

determine whether each statement is true or false. if the statement is false, make the necessary change(s) to produce a true statement.\and\ probabilities can always be determined using the formula p(a and b) = p(a) • p(b).choose the correct answer below. a. this statement is false. \and\ probabilities can only be determined using the formula p(a and b) = p(a) • p(b|a). b. this statement is true. c. this statement is false. \and\ probabilities can be determined using the formula p(a and b) = p(a) • p(b) only if a and b are independent. otherwise, they can be found using the formula p(a and b) = p(a) • p(b|a). d. this statement is false. \and\ probabilities can always be determined using the formula p(a and b) = p(a) + p(b).

Explanation:

The statement claims "And" probabilities can always use \(P(A \text{ and } B) = P(A) \cdot P(B)\). This is false because this formula only applies when events \(A\) and \(B\) are independent. For dependent events, the general formula is \(P(A \text{ and } B) = P(A) \cdot P(B|A)\). Option C correctly identifies the statement as false and explains the conditional use of the formulas.

Answer:

C. This statement is false. "And" probabilities can be determined using the formula \(P(A \text{ and } B) = P(A) \cdot P(B)\) only if A and B are independent. Otherwise, they can be found using the formula \(P(A \text{ and } B) = P(A) \cdot P(B|A)\).