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The two-way table shows the results of a recent study on the effectiveness of the flu vaccine. Let [tex]\( N \)[/tex] be the event that a person tested negative for the flu, and let [tex]\( V \)[/tex] be the event that the person was vaccinated.

\begin{tabular}{|c|c|c|c|}
\cline {2-4} \multicolumn{1}{c|}{} & Pos. & Neg. & Total \\
\hline Vaccinated & 465 & 771 & 1,236 \\
\hline \begin{tabular}{c}
Not \\
Vaccinated
\end{tabular} & 485 & 600 & 1,085 \\
\hline Total & 950 & 1,371 & 2,321 \\
\hline
\end{tabular}

Answer the questions to determine if events [tex]\( N \)[/tex] and [tex]\( V \)[/tex] are independent. Round your answers to the nearest hundredth.

[tex]\[ P(N \mid V ) = \square \][/tex]
[tex]\[ P(N) = \square \][/tex]

Are events [tex]\( N \)[/tex] and [tex]\( V \)[/tex] independent events? Yes or no?



Answer :

GinnyAnswer
To determine if the events [tex]\( N \)[/tex] (testing negative for the flu) and [tex]\( V \)[/tex] (being vaccinated) are independent, we need to calculate the following probabilities:
1. [tex]\( P(N \mid V) \)[/tex]: The probability of testing negative given that a person is vaccinated.
2. [tex]\( P(N) \)[/tex]: The overall probability of testing negative.

We'll start by computing these probabilities step-by-step using the information provided in the two-way table.

### Step 1: Calculate [tex]\( P(N \mid V) \)[/tex]
[tex]\( P(N \mid V) \)[/tex] is the conditional probability that a person tested negative given that they were vaccinated.

To compute [tex]\( P(N \mid V) \)[/tex]:
[tex]\[ P(N \mid V) = \frac{\text{Number of vaccinated people who tested negative}}{\text{Total number of vaccinated people}} \][/tex]

From the table:
- Number of vaccinated people who tested negative (Negative test among vaccinated) = 771
- Total number of vaccinated people (Vaccinated) = 1236

Thus:
[tex]\[ P(N \mid V) = \frac{771}{1236} \][/tex]

Rounding the result to the nearest hundredth:
[tex]\[ P(N \mid V) = 0.62 \][/tex]

### Step 2: Calculate [tex]\( P(N) \)[/tex]
[tex]\( P(N) \)[/tex] is the overall probability that a person tested negative, regardless of vaccination status.

To compute [tex]\( P(N) \)[/tex]:
[tex]\[ P(N) = \frac{\text{Number of people who tested negative}}{\text{Total number of people}} \][/tex]

From the table:
- Number of people who tested negative (Total negative cases) = 1371
- Total number of people (Total) = 2321

Thus:
[tex]\[ P(N) = \frac{1371}{2321} \][/tex]

Rounding the result to the nearest hundredth:
[tex]\[ P(N) = 0.59 \][/tex]

### Step 3: Determine Independence
Events [tex]\( N \)[/tex] and [tex]\( V \)[/tex] are independent if:
[tex]\[ P(N \mid V) = P(N) \][/tex]

Comparing the two probabilities we calculated:
- [tex]\( P(N \mid V) = 0.62 \)[/tex]
- [tex]\( P(N) = 0.59 \)[/tex]

Since [tex]\( P(N \mid V) \neq P(N) \)[/tex], events [tex]\( N \)[/tex] and [tex]\( V \)[/tex] are not independent.

### Final 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? No

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