Question

choose the three formulas that can be used to describe complementary events. select the three formulas that can be used to describe complementary events. a. ( p(e)=\frac{1}{p(e)} ) b. ( \frac{p(e)}{p(e)}=1 ) c. ( p(e) = 1 - p(e) ) d. ( p(e)=1 - p(e) ) e. ( p(e)=\frac{1}{p(e)} ) f. ( p(e)+p(e) = 1 ) g. ( p(e)-p(e) = 1 )

Explanation:

Step1: Recall the definition of complementary events

Complementary events: \(E\) and \(E'\) (where \(E'\) is the complement of \(E\)) satisfy the property that either \(E\) occurs or \(E'\) occurs, and there is no overlap. The sum of their probabilities is \(1\), i.e., \(P(E)+P(E') = 1\).

Step2: Derive other formulas from \(P(E)+P(E') = 1\)

If \(P(E)+P(E') = 1\), then by subtracting \(P(E)\) from both sides, we get \(P(E')=1 - P(E)\).
By subtracting \(P(E')\) from both sides of \(P(E)+P(E') = 1\), we get \(P(E)=1 - P(E')\).

Answer:

C. \(P(E') = 1 - P(E)\), D. \(P(E)=1 - P(E')\), F. \(P(E)+P(E') = 1\)