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)cdot p(b) )
choose the correct answer below
a. this statement is false. \and\ probabilities can be determined using the formula ( p(a and b)=p(a)cdot p(b) ) only if a and b are independent. otherwise, they can be
found using the formula ( p(a and b)=p(a)cdot p(b|a) )
b. this statement is false. \and\ probabilities can always be determined using the formula ( p(a and b)=p(a)+p(b) )
c. this statement is false. \and\ probabilities can only be determined using the formula ( p(a and b)=p(a)cdot p(b|a) )
d. this statement is true

Explanation:

The formula \(P(A\ \text{and}\ B)=P(A)\cdot P(B)\) is only valid for independent events. For dependent events, the formula \(P(A\ \text{and}\ B)=P(A)\cdot P(B|A)\) is used. Option A correctly identifies the conditions for using the two different "and" probability formulas. Option B is incorrect as \(P(A)+P(B)\) is the formula for the probability of the union of two mutually - exclusive events (for "or" probabilities). Option C is incorrect because it ignores the case of independent events where \(P(A\ \text{and}\ B)=P(A)\cdot P(B)\) (since \(P(B|A) = P(B)\) when \(A\) and \(B\) are independent).

Answer:

A. 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 \(P(A\ \text{and}\ B)=P(A)\cdot P(B|A)\).