Expected number affected = 2000 × (1/1000) = 2.

Expected number of affected people = (50,000 / 1000) = 50.

Expected number of affected people = (1/1000) × 10,000 = 10.

Expected affected = (1/200) × 1000 = 5 people.

Expected transmissions = 10% of 50 = 0.1 * 50 = 5 transmissions.

Potential infections = R0^n = 2^3 = 8.

If R0 = 3, then 1 infected person can infect 3 others. Therefore, potential infections = 3 * 1 = 3 people.

If the mortality rate is 10%, then 90% will survive. Therefore, expected survivors = 1000 × 0.9 = 900.

Expected survivors = Total infected - (Mortality rate × Total infected) = 500 - (0.10 × 500) = 500 - 50 = 450.

Expected survivors = Total infected - (Mortality rate × Total infected) = 200 - (0.05 × 200) = 200 - 10 = 190.

Expected recoveries = 85% of 200 = 0.85 × 200 = 170.

Expected recoveries = Recovery rate × Total patients = 0.9 × 50 = 45.

If the vaccine is 80% effective, then 20% will still contract the disease. Therefore, expected cases = 20% of 1000 = 200.

Expected cases = (1/1000) * 50000 = 50 cases.

If R0 = 3, one infected person can potentially infect 3 other people in a population without immunity.

Probability of not transmitting in one contact = 1 - 0.1 = 0.9. For 3 contacts, it is 0.9^3 = 0.729.

The probability of a false positive is equal to 1 - specificity = 1 - 0.95 = 0.05.

The probability that the disease will not be transmitted = 1 - 0.3 = 0.7.

The probability of a correct diagnosis = 1 - probability of misdiagnosis = 1 - 0.05 = 0.95.

Total infections = R0^n = 2.5^3 = 15.625.

Expected deaths = Case fatality rate × Number of cases = 0.04 × 250 = 10.

Expected new cases = 0.1% of 1,000,000 = 0.001 * 1,000,000 = 100 cases.

Total infected after 3 weeks = 10% + 10% of (100% - 10%) + 10% of (100% - 20%) = 10% + 9% + 8% = 27%.

Using the formula for compound growth: Total infected = 100 × (1 - (1 - 0.03)^4) = 100 × (1 - 0.88) = 12.36%.

Probability of contracting and not recovering = Probability of contracting × (1 - Probability of recovery) = 0.20 × (1 - 0.30) = 0.20 × 0.70 = 0.14.

Expected affected individuals = Prevalence × Population = 0.03 × 1000 = 30.

True positive rate = Sensitivity × Prevalence = 0.9 × 0.01 = 0.009.

Using Bayes' theorem, P(Disease | Positive) = (Sensitivity × Prevalence) / ((Sensitivity × Prevalence) + (1 - Specificity) × (1 - Prevalence)) = (0.9 × 0.01) / ((0.9 × 0.01) + (0.05 × 0.99)) = 0.018.

Using the binomial probability formula: P(X=k) = C(n,k) * p^k * (1-p)^(n-k). Here, n=5, k=3, p=0.8. Thus, P(3 out of 5) = C(5,3) * (0.8)^3 * (0.2)^2 = 10 * 0.512 * 0.04 = 0.2048.

Probability that treatment is not effective for one patient = 1 - 0.8 = 0.2. For three patients, the probability that it is not effective for all = 0.2^3 = 0.008. Therefore, the probability that it is effective for at least one = 1 - 0.008 = 0.992.