Question

determining if events are independent

this week in school, there is a 75 percent probability of having a fire drill, a 50 percent probability of a tornado drill, and a 25 percent probability of having both drills. let event f be a fire drill and event t be a tornado drill. are the two events independent?

no, because \\(p(f \cap t) \
eq p(f) \cdot p(t)\\)
no, because \\(p(f \cap t) \
eq p(f) + p(t)\\)
yes, because \\(p(f \cap t) = p(f) \cdot p(t)\\)
yes, because \\(p(f) = p(t) + p(f \cap t)\\)

Explanation:

Identify the given probabilities

Using the Independent Events and Probability Multiplication Rule knowledge points

$$
LATEXBLOCK0
$$

Calculate the product of individual probabilities

Using the Probability Multiplication Rule knowledge point

$$ P(F) \cdot P(T) = 0.75 \cdot 0.50 = 0.375 $$

Compare the intersection with the product

Using the Independent Events knowledge point

$$
LATEXBLOCK1
$$

Since the actual joint probability does not equal the product of their individual probabilities, the events are not independent.

Answer:

• **(A) No, because \(P(F \cap T)

eq P(F) \cdot P(T)\) (Correct answer)**

• (B) No, because \(P(F \cap T)

eq P(F) + P(T)\)

• (C) Yes, because \(P(F \cap T) = P(F) \cdot P(T)\)
• (D) Yes, because \(P(F) = P(T) + P(F \cap T)\)