Elsa must travel to a foreign country where a deadly strain of flu has ravaged the population. Already, 30%  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 5 out of every 16 people have been vaccinated. Thirty-five percent of those who are not vaccinated die from the flu. Elsa has just enough time to be vaccinated before she leaves. What is the probability that Elsa 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.) (Use only the law of total probabilities to solve this problem)

To solve this problem using the **law of total probability**, we will break it down step by step.

### Step 1: Define the events

- Let \( V \) be the event that Elsa is vaccinated.
- Let \( \neg V \) be the event that Elsa is not vaccinated.
- Let \( D \) be the event that Elsa dies from the flu.

We are tasked with finding \( P(D) \), the probability that Elsa will succumb to the flu and die.

According to the law of total probability, we can calculate \( P(D) \) as:

\[
P(D) = P(D|V)P(V) + P(D|\neg V)P(\neg V)
\]

Where:
- \( P(D|V) \) is the probability of dying given that Elsa is vaccinated.
- \( P(D|\neg V) \) is the probability of dying given that Elsa is not vaccinated.
- \( P(V) \) is the probability of being vaccinated.
- \( P(\neg V) \) is the probability of not being vaccinated.

### Step 2: Input known values

- \( P(V) \): The probability that Elsa is vaccinated is given as 5 out of 16 people, so:
  \[
  P(V) = \frac{5}{16}
  \]

- \( P(\neg V) \): The probability that Elsa is not vaccinated is the complement of \( P(V) \), so:
  \[
  P(\neg V) = 1 - P(V) = 1 - \frac{5}{16} = \frac{11}{16}
  \]

- \( P(D|V) \): Since the problem does not indicate that vaccinated people die from the flu, we assume the vaccine is completely effective. Therefore:
  \[
  P(D|V) = 0
  \]

- \( P(D|\neg V) \): The probability that someone who is not vaccinated dies from the flu is given as 35%, or:
  \[
  P(D|\neg V) = 0.35
  \]

### Step 3: Calculate \( P(D) \)

Substituting the known values into the law of total probability formula:

\[
P(D) = P(D|V)P(V) + P(D|\neg V)P(\neg V)
\]
\[
P(D) = (0)(\frac{5}{16}) + (0.35)(\frac{11}{16})
\]
\[
P(D) = 0 + 0.35 \times \frac{11}{16}
\]
\[
P(D) = \frac{0.35 \times 11}{16} = \frac{3.85}{16} \approx 0.24
\]

### Final Answer:
The probability that Elsa will succumb to the flu and die while she is away is approximately **0.24** or **24%**.

---

Do you want further details or clarifications on any of the steps?

Here are five related questions:
1. How would the probability change if the effectiveness of the vaccine was not 100%?
2. What if the percentage of vaccinated individuals increased?
3. How would this probability differ if the flu's mortality rate for unvaccinated individuals was higher?
4. What is the general form of the law of total probability?
5. How can conditional probabilities help in assessing medical risks?

**Tip**: The law of total probability is a great tool for calculating outcomes over mutually exclusive events, like whether someone is vaccinated or not.

Elsa must travel to a foreign country where a deadly strain of flu has ravaged the population. Already, 30% 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 5 out of every 16 people have been vaccinated. Thirty-five percent of those who are not vaccinated die from the flu. Elsa has just enough time to be vaccinated before she leaves. What is the probability that Elsa will succumb to the flu and die while she is away? Round your answer to two decimal places.

Find out the probability that Elsa will die from the flu using the law of total probability. This problem covers the core concept of complementary events and the calculation of probabilities in real-life health scenarios. The flu has a 35% mortality rate for the unvaccinated, and 5 out of 16 people are vaccinated. Suitable for students in Grades 10-12.

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