Question

determine whether the following statement is true or false. if it is false, explain why.

the probability that event a or event b will occur is \\(p(a \text{ or } b) = p(a) + p(b) + p(a \text{ and } b)\\).

choose the correct answer below.

a. false, the probability that a or b will occur is \\(p(a \text{ or } b) = p(a) + p(b) - p(a \text{ and } b)\\).
b. false, the probability that a or b will occur is \\(p(a \text{ or } b) = p(a) \cdot p(b)\\).
c. true
d. false, the probability that a or b will occur is \\(p(a \text{ or } b) = p(a) + p(b)\\).

Explanation:

Analyze the given statement

Using the Compound Event knowledge point, we identify that the statement describes the probability of the union of two events, \(A\) or \(B\). The statement claims:

$$ P(A \text{ or } B) = P(A) + P(B) + P(A \text{ and } B) $$

Apply the Addition Rule

Using the Probability Axioms knowledge point

$$ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) $$

Evaluate the options

Comparing our correct formula to the given choices:

• The original statement is false because it adds \(P(A \text{ and } B)\) instead of subtracting it.
• Option A correctly states that the statement is false and provides the correct subtraction formula.

Answer:

• (A) False, the probability that A or B will occur is P(A or B) = P(A) + P(B) - P(A and B). (Correct answer)
• (B) False, the probability that A or B will occur is P(A or B) = P(A) • P(B).
• (C) True
• (D) False, the probability that A or B will occur is P(A or B) = P(A) + P(B).