This problem can be approached using the binomial probability formula, which is:

$P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$

where:

• $n = 11$ (the number of patients),
• $k$ is the number of patients we want to find probabilities for (8 and 9, in this case),
• $p = 0.81$ (the probability of a single patient being cured),
• $\binom{n}{k}$ is the binomial coefficient, which calculates the number of ways to choose $k$ cured patients out of $n$ total patients.

To solve the problem, we need to calculate $P(X = 8)$ and $P(X = 9)$, then add these probabilities together.

Step 1: Calculate $P(X = 8)$

1. Binomial coefficient: $\binom{11}{8} = \frac{11!}{8!(11 - 8)!} = \frac{11!}{8! \cdot 3!} = 165$
2. Probability term for $k = 8$: $P(X = 8) = \binom{11}{8} (0.81)^8 (0.19)^3$ $P(X = 8) = 165 \cdot (0.81)^8 \cdot (0.19)^3$

Step 2: Calculate $P(X = 9)$

1. Binomial coefficient: $\binom{11}{9} = \frac{11!}{9!(11 - 9)!} = \frac{11!}{9! \cdot 2!} = 55$
2. Probability term for $k = 9$: $P(X = 9) = \binom{11}{9} (0.81)^9 (0.19)^2$ $P(X = 9) = 55 \cdot (0.81)^9 \cdot (0.19)^2$

Step 3: Add $P(X = 8)$ and $P(X = 9)$

Now, calculate both probabilities and add them up:

$P(8 \text{ or } 9 \text{ cured}) = P(X = 8) + P(X = 9)$

After performing these calculations (using exact values for accuracy in intermediate steps), round the final answer to three decimal places.

Let's calculate it fully:

Calculation:

1. $P(X = 8) = 165 \cdot (0.81)^8 \cdot (0.19)^3 \approx 0.233$
2. $P(X = 9) = 55 \cdot (0.81)^9 \cdot (0.19)^2 \approx 0.274$

$P(8 \text{ or } 9 \text{ cured}) \approx 0.233 + 0.274 = 0.507$

Final Answer:

The probability that exactly 8 or 9 out of the 11 patients are cured is approximately:

$\boxed{0.507}$

Would you like further details or have questions about any steps?

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1. What is a binomial probability distribution?
2. How is the binomial coefficient calculated?
3. What does it mean when we add probabilities for two values?
4. Can this process be applied to other probabilities?
5. What would happen if we increased the probability of success to 90%?

Tip: In binomial problems, always check if events are independent and only have two outcomes per trial.