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 the multiplication rule for independent events. For dependent events, the formula is \(P(A\ \text{and}\ B)=P(A)\cdot P(B|A)\). The statement in the question claims that "\(And\)" probabilities can always use \(P(A\ \text{and}\ B)=P(A)\cdot P(B)\), which is incorrect as it ignores the case of dependent events. Option A correctly identifies the error and provides the correct formulas for both independent and dependent cases. Option B is wrong because \(P(A\ \text{and}\ B)=P(A)+P(B)\) is the addition rule for mutually - exclusive events (which is for \(Or\) probabilities). Option C is wrong because it says "\(And\)" probabilities can only use \(P(A\ \text{and}\ B)=P(A)\cdot P(B|A)\), ignoring the valid use of \(P(A\ \text{and}\ B)=P(A)\cdot P(B)\) for independent events. Option D is wrong as the original statement is false.

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