Question

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:

Step1: Recall Independent Events Formula

For two independent events \( F \) and \( T \), the probability of their intersection is \( P(F \cap T)=P(F)\cdot P(T) \).

Step2: Calculate \( P(F)\cdot P(T) \)

Given \( P(F) = 0.75 \) (75%), \( P(T)=0.50 \) (50%). So \( P(F)\cdot P(T)=0.75\times0.50 = 0.375 \) (37.5%).

Step3: Compare with \( P(F \cap T) \)

Given \( P(F \cap T)=0.25 \) (25%). Since \( 0.25
eq0.375 \), i.e., \( P(F \cap T)
eq P(F)\cdot P(T) \), the events are not independent.

Answer:

No, because \( P(F \cap T)
eq P(F)\cdot P(T) \) (the first option).