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To solve these problems, we can model the situation using the binomial probability distribution, since we are dealing with a fixed number of trials (10 appeals), each with two possible outcomes (successful or not), and a constant probability of success (45%).

The probability mass function for a binomial distribution is given by:

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

where:
- \( n \) is the number of trials,
- \( k \) is the number of successful trials,
- \( p \) is the probability of success (0.45), and
- \( \binom{n}{k} \) is the binomial coefficient.

### Part (c): Probability that at least two appeals will be successful
We need to calculate \( P(X \geq 2) \). This can be done by finding the complement:

\[
P(X \geq 2) = 1 - P(X < 2) = 1 - [P(X = 0) + P(X = 1)]
\]

Using the given probabilities:
- \( P(X = 0) = 0.0025 \)
- \( P(X = 1) = 0.0207 \)

Thus,

\[
P(X \geq 2) = 1 - (0.0025 + 0.0207) = 1 - 0.0232 = 0.9768
\]

So, the probability that at least two appeals will be successful is 0.9768.

### Part (d): Probability that more than half of the appeals will be successful
This corresponds to \( P(X > 5) \), which means calculating the probabilities for \( X = 6, 7, 8, 9, \) and \( 10 \). We will need to use the binomial formula for each of these values or use a cumulative distribution table or software to get these probabilities.

Let me know if you'd like me to calculate it step by step or need further assistance.

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Would you like further details on this or have any questions?

### Related Questions:
1. How would the probabilities change if the probability of success was different, say 60%?
2. How can the binomial distribution be approximated using a normal distribution for large values of \( n \)?
3. What is the expected number of successful appeals out of 10?
4. How do we interpret the variance and standard deviation of a binomial distribution?
5. How would you solve this problem if appeals were not independent?

### Tip:
Always check if the problem can be simplified using complementary probabilities (like \( P(X \geq k) = 1 - P(X < k) \)), as it can make calculations much easier.

A center for medical services reported that there were 295,000 appeals for hospitalization and other services. For this group, 45% of first-round appeals were successful. Suppose 10 first-round appeals have just been received by a Medicare appeals office. (Round your answers to four decimal places.)

(a) Compute the probability that none of the appeals will be successful.
(b) Compute the probability that exactly one of the appeals will be successful.
(c) What is the probability that at least two of the appeals will be successful?
(d) What is the probability that more than half of the appeals will be successful?

Learn how to calculate the probability of successful first-round Medicare appeals using the binomial distribution. This includes finding the probability of none, exactly one, at least two, and more than half of the appeals being successful, given a 45% success rate. Suitable for Grades 11-12.

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