Let's work through the problem step by step:
1. Calculate [tex]\( P(N \mid V) \)[/tex]:
- [tex]\( P(N \mid V) \)[/tex] is the probability of a person testing negative for the flu given that they were vaccinated.
- From the table, the number of vaccinated people (event [tex]\( V \)[/tex]) is [tex]\( 1236 \)[/tex].
- The number of vaccinated people who tested negative for the flu is [tex]\( 771 \)[/tex].
Therefore:
[tex]\[
P(N \mid V) = \frac{\text{Number of vaccinated people who tested negative}}{\text{Number of vaccinated people}} = \frac{771}{1236}
\][/tex]
Using given results:
[tex]\[
P(N \mid V) = 0.62
\][/tex]
2. Calculate [tex]\( P(N) \)[/tex]:
- [tex]\( P(N) \)[/tex] is the overall probability of a person testing negative for the flu.
- From the table, the total population is [tex]\( 2321 \)[/tex].
- The total number of people who tested negative for the flu is [tex]\( 1371 \)[/tex].
Therefore:
[tex]\[
P(N) = \frac{\text{Number of people who tested negative}}{\text{Total population}} = \frac{1371}{2321}
\][/tex]
Using given results:
[tex]\[
P(N) = 0.59
\][/tex]
3. Determine if events [tex]\( N \)[/tex] and [tex]\( V \)[/tex] are independent:
- Events [tex]\( N \)[/tex] and [tex]\( V \)[/tex] are independent if and only if [tex]\( P(N \mid V) = P(N) \)[/tex].
Comparing:
[tex]\[
P(N \mid V) = 0.62
\][/tex]
[tex]\[
P(N) = 0.59
\][/tex]
Since [tex]\( 0.62 \neq 0.59 \)[/tex]:
[tex]\[
\text{Events } N \text{ and } V \text{ are not independent.}
\][/tex]
4. Provide the final answers:
[tex]\[
P(N \mid V) = 0.62
\][/tex]
[tex]\[
P(N) = 0.59
\][/tex]
[tex]\[
\text{Are events } N \text{ and } V \text{ independent? No.}
\][/tex]
