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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 [tex]$F$[/tex] be a fire drill and event [tex]$T$[/tex] be a tornado drill. Are the two events independent?

A. No, because [tex]$P(F \cap T ) \neq P(F) \cdot P(T)$[/tex].

B. No, because [tex][tex]$P(F \cap T ) \neq P(F)+P(T)$[/tex][/tex].

C. Yes, because [tex]$P(F \cap T ) = P(F) \cdot P(T)$[/tex].

D. Yes, because [tex]$P(F)= P (T) + P(F \cap T)$[/tex].

Sagot :

GinnyAnsweridn
To determine whether two events are independent, we need to check if the probability of both events occurring simultaneously, [tex]\( P(F \cap T) \)[/tex], is equal to the product of the individual probabilities of each event, [tex]\( P(F) \cdot P(T) \)[/tex].

Let's go through the steps:

1. Identify the given probabilities:
- Probability of a fire drill, [tex]\( P(F) \)[/tex] = 0.75
- Probability of a tornado drill, [tex]\( P(T) \)[/tex] = 0.5
- Probability of both drills happening, [tex]\( P(F \cap T) \)[/tex] = 0.25

2. Calculate the product of the individual probabilities:
[tex]\[ P(F) \cdot P(T) = 0.75 \times 0.5 = 0.375 \][/tex]

3. Compare [tex]\( P(F \cap T) \)[/tex] with [tex]\( P(F) \cdot P(T) \)[/tex]:
- [tex]\( P(F \cap T) = 0.25 \)[/tex]
- [tex]\( P(F) \cdot P(T) = 0.375 \)[/tex]

4. Since [tex]\( P(F \cap T) \neq P(F) \cdot P(T) \)[/tex], the two events are not independent.

Therefore, the correct answer is:

No, because [tex]\( P(F \cap T) \neq P(F) \cdot P(T) \)[/tex].
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