Leila must travel to a foreign country where a deadly strain of flu has ravaged the population. Already, 22% of the population has succumbed to the flu and died. There is a vaccine, but because the flu is extremely contagious, it spreads quite rapidly, and to date only 10 out of every 16 people have been vaccinated. Thirty-two percent of those who are not vaccinated die from the flu. Leila has just enough time to be vaccinated before she leaves. What is the probability that Leila will succumb to the flu and die while she is away? Round your answer to two decimal places. (If necessary, consult a list of formulas.)


To solve this problem, we need to determine the probability that Leila will succumb to the flu and die, given that she has time to get vaccinated. We'll break this problem down into steps.

### Step 1: Calculate the probability of dying for vaccinated and unvaccinated people.
From the problem:
- 22% of the population has already died due to the flu.
- 10 out of 16 people have been vaccinated, meaning the probability that a person is vaccinated is \( P(\text{vaccinated}) = \frac{10}{16} = 0.625 \), and the probability that a person is unvaccinated is \( P(\text{not vaccinated}) = \frac{6}{16} = 0.375 \).
- 32% of unvaccinated people die from the flu, so the probability of dying if unvaccinated is \( P(\text{die | not vaccinated}) = 0.32 \).
- We assume that vaccinated people are fully protected, meaning the probability of dying if vaccinated is \( P(\text{die | vaccinated}) = 0 \).

### Step 2: Use the Law of Total Probability.
The probability that Leila dies depends on whether or not she is vaccinated. Since she is planning to get vaccinated, the probability of dying can be found by considering both the vaccinated and unvaccinated scenarios.

\[
P(\text{die}) = P(\text{vaccinated}) \times P(\text{die | vaccinated}) + P(\text{not vaccinated}) \times P(\text{die | not vaccinated})
\]

Substitute the known values:

\[
P(\text{die}) = 0.625 \times 0 + 0.375 \times 0.32
\]

\[
P(\text{die}) = 0 + 0.375 \times 0.32 = 0.12
\]

Thus, the probability that Leila will succumb to the flu and die is \( 0.12 \).

### Final Answer:
The probability that Leila will succumb to the flu and die is **0.12**, or **12%**.

Would you like further details, or do you have any questions?

Here are 5 related questions:
1. What is the probability Leila will survive if she does not get vaccinated?
2. How does the population death rate impact Leila’s risk if she doesn't vaccinate?
3. If only half of the population got vaccinated, what would the probability of death be for Leila?
4. What happens to the overall mortality rate if the flu becomes even more contagious?
5. How would the probability change if the flu mutates and becomes vaccine-resistant?

**Tip:** When solving probability problems, break the problem into smaller parts and use tools like the Law of Total Probability to handle conditional scenarios.

Leila must travel to a foreign country where a deadly strain of flu has ravaged the population. Already, 22% of the population has succumbed to the flu and died. There is a vaccine, but because the flu is extremely contagious, it spreads quite rapidly, and to date only 10 out of every 16 people have been vaccinated. Thirty-two percent of those who are not vaccinated die from the flu. Leila has just enough time to be vaccinated before she leaves. What is the probability that Leila will succumb to the flu and die while she is away? Round your answer to two decimal places. (If necessary, consult a list of formulas.)

Determine the probability that Leila will succumb to the flu given that she plans to be vaccinated. Understand the impact of vaccination on survival rates using the Law of Total Probability. The problem includes key concepts like conditional probability and is suitable for students in Grades 10-12.

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