```latex
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Age & Under 15 & [tex]$15-24$[/tex] & [tex]$25-44$[/tex] & [tex]$45-64$[/tex] & 65 and older \\
\hline
Proportion & [tex]$20 \%$[/tex] & [tex]$10 \%$[/tex] & [tex]$25 \%$[/tex] & [tex]$20 \%$[/tex] & [tex]$25 \%$[/tex] \\
\hline
\end{tabular}

Suppose you are a district manager of a health management organization (HMO) that is monitoring the office of a local doctor or nurse in general family practice. This morning the office you are monitoring has eight office visits on the schedule. What is the probability of the following?

(a) At least half the patients are under 15 years old. (Round your answer to three decimal places.)
[tex]$\square$[/tex]
Explain how this can be modeled as a binomial distribution with 8 trials, where success is visitor age under 15 years old and the probability of success is [tex]$20\%$[/tex].

Let [tex]$n=8, p=0.20$[/tex] and compute the probabilities using the binomial distribution.

(b) From 2 to 5 patients are 65 years old or older. (Round your answer to three decimal places.)
[tex]$\square$[/tex]

(c) From 2 to 5 patients are 45 years old or older. (Hint: Success if 45 or older. Use the table to compute the probability of success on a single visit. Round your answer to three decimal places.)
[tex]$\square$[/tex]

(d) All the patients are under 25 years of age. (Round your answer to three decimal places.)
[tex]$\square$[/tex]
```



Answer :

Absolutely, let's go through this problem step-by-step:

### Part (a): At least half the patients are under 15 years old

1. Define the variables:
- Number of trials, [tex]\( n = 8 \)[/tex] (the number of office visits).
- Probability of success (a patient being under 15 years old), [tex]\( p = 0.20 \)[/tex].

2. Identify the requirement:
- We need to calculate the probability that at least half (4 out of 8) of the patients are under 15 years old.

3. Apply the binomial distribution:
- We need to find the probability for [tex]\( X \geq 4 \)[/tex] where [tex]\( X \)[/tex] is the number of patients under 15 years old.
- [tex]\( P(X \geq 4) = 1 - P(X < 4) \)[/tex].

4. Calculate [tex]\( P(X < 4) \)[/tex] using the binomial distribution:
- Sum the individual probabilities for [tex]\( X = 0, 1, 2, 3 \)[/tex].

Finally, the calculated probability will be approximately [tex]\( \boxed{0.056} \)[/tex].

### Part (b): From 2 to 5 patients are 65 years old or older

1. Define the variables:
- Number of trials, [tex]\( n = 8 \)[/tex].
- Probability of success (a patient being 65 years or older), [tex]\( p = 0.25 \)[/tex].

2. Identify the requirement:
- We need to calculate the probability that the number of patients who are 65 years old or older is between 2 and 5, inclusive.

3. Apply the binomial distribution:
- We need to find the probability for [tex]\( 2 \leq X \leq 5 \)[/tex] where [tex]\( X \)[/tex] is the number of patients 65 years or older.

4. Calculate the probabilities:
- Sum the probabilities for [tex]\( X = 2, 3, 4, 5 \)[/tex].

The resulting probability would be approximately [tex]\( \boxed{0.629} \)[/tex].

### Part (c): From 2 to 5 patients are 45 years old or older

1. Define the variables:
- Number of trials, [tex]\( n = 8 \)[/tex].
- Probability of success (a patient being 45 years or older), [tex]\( p = 0.20 + 0.25 = 0.45 \)[/tex].

2. Identify the requirement:
- We need to calculate the probability that the number of patients who are 45 years old or older is between 2 and 5, inclusive.

3. Apply the binomial distribution:
- We need to find the probability for [tex]\( 2 \leq X \leq 5 \)[/tex] where [tex]\( X \)[/tex] is the number of patients 45 years or older.

4. Calculate the probabilities:
- Sum the probabilities for [tex]\( X = 2, 3, 4, 5 \)[/tex].

The resulting probability would be approximately [tex]\( \boxed{0.848} \)[/tex].

### Part (d): All the patients are under 25 years of age

1. Define the variables:
- Number of trials, [tex]\( n = 8 \)[/tex].
- Probability of success (a patient being under 25 years old), [tex]\( p = 0.20 + 0.10 = 0.30 \)[/tex].

2. Identify the requirement:
- We need to calculate the probability that all patients (8 out of 8) are under 25 years old.

3. Apply the binomial distribution:
- We need to find the probability for [tex]\( X = 8 \)[/tex] where [tex]\( X \)[/tex] is the number of patients under 25 years old.

4. Calculate the probability:
- [tex]\( P(X = 8) \)[/tex].

The resulting probability would be approximately [tex]\( \boxed{0.000} \)[/tex].

Each of these probabilities encapsulates the likelihood of the respective events happening under the conditions given.