To determine if events [tex]\(N\)[/tex] (testing negative for the flu) and [tex]\(V\)[/tex] (being vaccinated) are independent, we need to compute the following probabilities and compare them:
1. [tex]\(P(N \mid V)\)[/tex]: The probability of testing negative given that a person was vaccinated.
2. [tex]\(P(N)\)[/tex]: The probability of testing negative in the overall population.
Step 1: Calculate [tex]\(P(N \mid V)\)[/tex]
This is the probability of a person testing negative given that they were vaccinated. Using the table provided:
- Number of vaccinated individuals who tested negative ([tex]\( N \)[/tex] given [tex]\( V \)[/tex]): 771
- Total number of vaccinated individuals ([tex]\( V \)[/tex]): 1236
[tex]\[ P(N \mid V) = \frac{\text{Number of vaccinated individuals who tested negative}}{\text{Total number of vaccinated individuals}} = \frac{771}{1236} \][/tex]
When rounded to the nearest hundredth:
[tex]\[ P(N \mid V) \approx 0.62 \][/tex]
Step 2: Calculate [tex]\(P(N)\)[/tex]
This is the probability of a person testing negative in the overall population. Using the table provided:
- Total number of individuals who tested negative ([tex]\( N \)[/tex]): 1371
- Total population: 2321
[tex]\[ P(N) = \frac{\text{Total number of individuals who tested negative}}{\text{Total population}} = \frac{1371}{2321} \][/tex]
When rounded to the nearest hundredth:
[tex]\[ P(N) \approx 0.59 \][/tex]
Step 3: Compare [tex]\( P(N \mid V) \)[/tex] and [tex]\( P(N) \)[/tex] to check for independence
Events [tex]\(N\)[/tex] and [tex]\(V\)[/tex] are considered independent if and only if [tex]\( P(N \mid V) = P(N) \)[/tex].
Given that:
[tex]\[ P(N \mid V) \approx 0.62 \][/tex]
[tex]\[ P(N) \approx 0.59 \][/tex]
Since [tex]\( 0.62 \neq 0.59 \)[/tex], the events [tex]\(N\)[/tex] and [tex]\(V\)[/tex] are not independent.
Answers:
[tex]\[ P(N \mid V) = 0.62 \][/tex]
[tex]\[ P(N) = 0.59 \][/tex]
Are events [tex]\(N\)[/tex] and [tex]\(V\)[/tex] independent events?
No.
