determine whether each statement is true or false. if the statement is false, make the necessary change(s)…

determine whether each statement is true or false. if the statement is false, make the necessary change(s) to produce a true statement\n\and\ probabilities can always be determined using the formula ( p(a and b)=p(a)cdot p(b) )\nchoose the correct answer below\na. 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\nfound using the formula ( p(a and b)=p(a)cdot p(b|a) )\nb. this statement is false. \and\ probabilities can always be determined using the formula ( p(a and b)=p(a)+p(b) )\nc. this statement is false. \and\ probabilities can only be determined using the formula ( p(a and b)=p(a)cdot p(b|a) )\nd. this statement is true

determine whether each statement is true or false. if the statement is false, make the necessary change(s) to produce a true statement\n\and\ probabilities can always be determined using the formula ( p(a and b)=p(a)cdot p(b) )\nchoose the correct answer below\na. 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\nfound using the formula ( p(a and b)=p(a)cdot p(b|a) )\nb. this statement is false. \and\ probabilities can always be determined using the formula ( p(a and b)=p(a)+p(b) )\nc. this statement is false. \and\ probabilities can only be determined using the formula ( p(a and b)=p(a)cdot p(b|a) )\nd. this statement is true

Answer

Brief Explanations:

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)).