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.
