Question

fill in the blank so that the resulting statement is true. if it is possible for the events a and b to occur simultaneously, then p(a or b) = ______. if it is possible for the events a and b to occur simultaneously, then p(a or b) = dropdown with options: p(a and b), p(a) + p(b) - p(a and b), 1 - p(a and b), p(a) + p(b)

Explanation:

Step1: Recall the principle of inclusion - exclusion for probability.

When two events \(A\) and \(B\) can occur simultaneously (i.e., they are not mutually exclusive), the probability of \(A\) or \(B\) occurring is given by the formula that accounts for the overlap (the probability that both \(A\) and \(B\) occur) to avoid double - counting.
The formula for \(P(A\cup B)\) (which is the same as \(P(A\text{ or }B)\)) is derived from the fact that if we just add \(P(A)\) and \(P(B)\), we have counted the probability of \(A\) and \(B\) occurring together twice. So we need to subtract \(P(A\cap B)\) (which is \(P(A\text{ and }B)\)) once to get the correct probability.
Mathematically, \(P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)\).

Step2: Analyze the other options.

• Option " \(P(A\text{ and }B)\)": This is the probability that both \(A\) and \(B\) occur, not the probability that \(A\) or \(B\) occurs. So this option is incorrect.
• Option " \(1 - P(A\text{ and }B)\)": There is no basis for this formula in the context of finding \(P(A\text{ or }B)\). This formula does not relate to the probability of \(A\) or \(B\) occurring. So this option is incorrect.
• Option " \(P(A)+P(B)\)": This formula is used when \(A\) and \(B\) are mutually exclusive (i.e., they cannot occur simultaneously). When \(A\) and \(B\) can occur simultaneously, using \(P(A)+P(B)\) would double - count the region where \(A\) and \(B\) overlap. So this option is incorrect.

Answer:

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