To determine if the events [tex]\( N \)[/tex] and [tex]\( V \)[/tex] are independent, we need to calculate the following probabilities:
1. [tex]\( P(N \mid V) \)[/tex] - the probability of testing negative for the flu given that a person was vaccinated.
2. [tex]\( P(N) \)[/tex] - the overall probability of testing negative for the flu.
Let's break down the steps to find these probabilities:
### Step 1: Calculate [tex]\( P(N \mid V) \)[/tex]
The formula to calculate [tex]\( P(N \mid V) \)[/tex] is:
[tex]\[ P(N \mid V) = \frac{\text{Number of vaccinated people who tested negative}}{\text{Total number of vaccinated people}} \][/tex]
From the table, we have:
- Number of vaccinated people who tested negative = 771
- Total number of vaccinated people = 1236
So,
[tex]\[ P(N \mid V) = \frac{771}{1236} \][/tex]
### Step 2: Calculate [tex]\( P(N) \)[/tex]
The formula to calculate [tex]\( P(N) \)[/tex] is:
[tex]\[ P(N) = \frac{\text{Total number of people who tested negative}}{\text{Total number of people}} \][/tex]
From the table, we have:
- Total number of people who tested negative = 1371
- Total number of people = 2321
So,
[tex]\[ P(N) = \frac{1371}{2321} \][/tex]
### Step 3: Determine Independence
Two events, [tex]\( N \)[/tex] and [tex]\( V \)[/tex], are independent if:
[tex]\[ P(N \mid V) = P(N) \][/tex]
### Calculations
Now, we plug in the numbers and round to the nearest hundredth:
1. Calculate [tex]\( P(N \mid V) \)[/tex]:
[tex]\[ P(N \mid V) = \frac{771}{1236} \approx 0.62 \][/tex]
2. Calculate [tex]\( P(N) \)[/tex]:
[tex]\[ P(N) = \frac{1371}{2321} \approx 0.59 \][/tex]
### Conclusion
Since [tex]\( P(N \mid V) \approx 0.62 \)[/tex] and [tex]\( P(N) \approx 0.59 \)[/tex] are not equal, the events [tex]\( N \)[/tex] and [tex]\( V \)[/tex] are not independent.
### Summary of 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.
