5. One quarter of a population has been vaccinated against a contagious disease. During an epidemic, one
finds that there is, among the sick, one vaccinated person for every four unvaccinated. We know that during the
epidemic, there was one sick person in 12 among the vaccinated. What was the probability of becoming ill for an
unvaccinated individual? Is the vaccine effective?

To solve this problem, we need to apply the concepts of conditional probability and Bayes' theorem. We can break down the information provided and use it to calculate the probability of an unvaccinated individual becoming ill. We'll also determine whether the vaccine is effective.

### **Step 1: Define the Problem**

Let's denote:
- \( V \) = the event that a person is vaccinated.
- \( \overline{V} \) = the event that a person is not vaccinated.
- \( I \) = the event that a person becomes ill.
- \( \overline{I} \) = the event that a person does not become ill.

### **Step 2: Given Information**

- \( P(V) = \frac{1}{4} \) (since one-quarter of the population is vaccinated).
- \( P(\overline{V}) = \frac{3}{4} \) (since three-quarters of the population is unvaccinated).
- Among the sick, the ratio of vaccinated to unvaccinated individuals is 1:4.
- \( P(I|V) = \frac{1}{12} \) (the probability of illness given vaccination).

### **Step 3: Determine the Probability of Illness Among the Unvaccinated**

We need to find \( P(I|\overline{V}) \), the probability of an unvaccinated individual becoming ill.

We know the ratio of the number of sick vaccinated individuals to sick unvaccinated individuals is 1:4. This implies:

\[
\frac{N_{I \cap V}}{N_{I \cap \overline{V}}} = \frac{1}{4}
\]

Where \( N_{I \cap V} \) is the number of vaccinated sick individuals and \( N_{I \cap \overline{V}} \) is the number of unvaccinated sick individuals.

Using the law of total probability for the sick individuals \( N_I \):

\[
N_I = N_{I \cap V} + N_{I \cap \overline{V}}
\]

Given the ratio \( \frac{N_{I \cap V}}{N_{I \cap \overline{V}}} = \frac{1}{4} \), we know:

\[
N_I = N_{I \cap V} + 4N_{I \cap V} = 5N_{I \cap V}
\]

So,

\[
N_{I \cap \overline{V}} = 4N_{I \cap V}
\]

### **Step 4: Calculate the Total Probability of Being Sick**

Let \( P(I) \) be the total probability of being sick. We use:

\[
P(I) = P(V)P(I|V) + P(\overline{V})P(I|\overline{V})
\]

We can express the probability of being sick for the unvaccinated:

\[
P(I|\overline{V}) = \frac{4N_{I \cap V}}{N_{\overline{V}}}
\]

### **Step 5: Solve for \( P(I|\overline{V}) \)**

Since \( P(I|V) = \frac{1}{12} \) and \( P(V) = \frac{1}{4} \):

\[
P(V \cap I) = P(I|V) \times P(V) = \frac{1}{12} \times \frac{1}{4} = \frac{1}{48}
\]

Using the ratio and substituting into \( P(I|\overline{V}) \):

\[
P(I|\overline{V}) = \frac{4 \times \frac{1}{48}}{\frac{3}{4}} = \frac{4}{48} \times \frac{4}{3} = \frac{1}{12} \times \frac{4}{3} = \frac{1}{12} \times \frac{4}{3} = \frac{4}{36} = \frac{1}{9}
\]

So, the probability of becoming ill for an unvaccinated individual is \( P(I|\overline{V}) = \frac{1}{9} \).

### **Step 6: Is the Vaccine Effective?**

To determine if the vaccine is effective, compare the probabilities:

- \( P(I|V) = \frac{1}{12} \)
- \( P(I|\overline{V}) = \frac{1}{9} \)

Since \( \frac{1}{12} < \frac{1}{9} \), the vaccine reduces the likelihood of becoming ill. Therefore, **the vaccine is effective**.

Would you like further details on any part of this solution, or do you have more questions?

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**Related Questions:**
1. How does Bayes' theorem apply to this problem?
2. How can we generalize this approach to other vaccination effectiveness studies?
3. What assumptions are made in this calculation regarding the population and the vaccine?
4. How would the results change if the ratio of vaccinated to unvaccinated sick individuals was different?
5. How can we calculate the overall effectiveness of a vaccine using population data?

**Tip:** In real-world applications, vaccine effectiveness is often reported as a percentage reduction in disease incidence in vaccinated individuals compared to unvaccinated individuals.

Learn how to calculate the probability of illness for unvaccinated individuals using conditional probability and Bayes' Theorem. This problem analyzes vaccination effectiveness during an epidemic, applying mathematical concepts including conditional probability and Bayes' Theorem.

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