Based on data from a statistical abstract, only about 10% of senior citizens (65 years old or older) get the flu each year. However, about 27% of the people under 65 years old get the flu each year. In the general population, there are 15% senior citizens (65 years old or older). (Round your answers to three decimal places.)

(a) What is the probability that a person selected at random from the general population is senior citizen who will get the flu this season?

(b) What is the probability that a person selected at random from the general population is a person under age 65 who will get the flu this year?

(c) Repeat parts (a) and (b) for a community that has 92% senior citizens.

(d) Repeat parts (a) and (b) for a community that has 52% senior citizens.

a. .10 * .15 = ?

b. .27 * (1-.15) = ?

Use similar process for c and d.

Idk

To solve this problem, we can use the concept of conditional probability. Let's break down the steps for each part:

(a) To find the probability that a random person from the general population is a senior citizen who will get the flu this season, we need to find the intersection of these two events: being a senior citizen and getting the flu.

Let's denote:
A = being a senior citizen
B = getting the flu

From the given information:
P(A) = 0.15 (probability of being a senior citizen)
P(B|A) = 0.10 (probability of getting the flu given that the person is a senior citizen)

To find P(A ∩ B), we can multiply the probabilities:
P(A ∩ B) = P(A) * P(B|A)

Substituting the values:
P(A ∩ B) = 0.15 * 0.10 = 0.015

Therefore, the probability that a random person from the general population is a senior citizen who will get the flu this season is 0.015.

(b) Similarly, to find the probability that a random person from the general population is a person under age 65 who will get the flu this year, we need to find the intersection of these two events: being under age 65 and getting the flu.

Let's denote:
C = being under age 65
D = getting the flu

From the given information:
P(C) = 1 - P(A) = 1 - 0.15 (probability of being under age 65)
P(D|C) = 0.27 (probability of getting the flu given that the person is under age 65)

To find P(C ∩ D), we can multiply the probabilities:
P(C ∩ D) = P(C) * P(D|C)

Substituting the values:
P(C ∩ D) = (1 - 0.15) * 0.27 = 0.85 * 0.27 = 0.2295

Therefore, the probability that a random person from the general population is a person under age 65 who will get the flu this year is 0.2295.

(c) Let's repeat the calculations for a community that has 92% senior citizens.

For part (a):
P(A) = 0.92
P(B|A) = 0.10

P(A ∩ B) = P(A) * P(B|A) = 0.92 * 0.10 = 0.092

Therefore, the probability in this community is 0.092.

For part (b):
P(C) = 1 - P(A) = 1 - 0.92 = 0.08
P(D|C) = 0.27

P(C ∩ D) = P(C) * P(D|C) = 0.08 * 0.27 = 0.0216

Therefore, the probability in this community is 0.0216.

(d) Let's repeat the calculations for a community that has 52% senior citizens.

For part (a):
P(A) = 0.52
P(B|A) = 0.10

P(A ∩ B) = P(A) * P(B|A) = 0.52 * 0.10 = 0.052

Therefore, the probability in this community is 0.052.

For part (b):
P(C) = 1 - P(A) = 1 - 0.52 = 0.48
P(D|C) = 0.27

P(C ∩ D) = P(C) * P(D|C) = 0.48 * 0.27 = 0.1296

Therefore, the probability in this community is 0.1296.

To find the probabilities, we need to use the given information and apply some basic statistics concepts.

(a) Let's find the probability that a person selected at random from the general population is a senior citizen who will get the flu this season.
We know that only 10% of senior citizens get the flu. The general population has a 15% of senior citizens. First, we need to calculate the probability that a randomly chosen person from the general population is a senior citizen:

P(Senior Citizen) = 15% = 0.15

Once we know the person is a senior citizen, the probability of getting the flu is 10%:

P(Flu | Senior Citizen) = 10% = 0.10

To find the probability that a person is both a senior citizen and will get the flu, we multiply the two probabilities:

P(Senior Citizen and Flu) = P(Senior Citizen) * P(Flu | Senior Citizen)
= 0.15 * 0.10
= 0.015

Therefore, the probability that a person selected at random from the general population is a senior citizen who will get the flu this season is 0.015 (or 1.5%).

(b) Now let's find the probability that a person selected at random from the general population is a person under age 65 who will get the flu this year. We know that 27% of people under 65 years old get the flu.
We already calculated the probability that a person is a senior citizen (P(Senior Citizen) = 0.15). To find the probability that a person is under 65, we subtract the probability of being a senior citizen from 1:

P(Under 65) = 1 - P(Senior Citizen)
= 1 - 0.15
= 0.85

To find the probability that a person is both under 65 and will get the flu, we multiply the two probabilities:

P(Under 65 and Flu) = P(Under 65) * P(Flu | Under 65)
= 0.85 * 0.27
= 0.2295

Therefore, the probability that a person selected at random from the general population is a person under age 65 who will get the flu this year is 0.2295 (or 22.95%).

(c) To repeat parts (a) and (b) for a community that has 92% senior citizens, we use the same method, but with different probabilities.

For part (a):
P(Senior Citizen) = 92% = 0.92
P(Flu | Senior Citizen) = 10% = 0.10

P(Senior Citizen and Flu) = P(Senior Citizen) * P(Flu | Senior Citizen)
= 0.92 * 0.10
= 0.092

Therefore, the probability that a person selected at random from a community with 92% senior citizens is a senior citizen who will get the flu this season is 0.092 (or 9.2%).

For part (b):
P(Under 65) = 1 - P(Senior Citizen)
= 1 - 0.92
= 0.08

P(Under 65 and Flu) = P(Under 65) * P(Flu | Under 65)
= 0.08 * 0.27
= 0.0216

Therefore, the probability that a person selected at random from a community with 92% senior citizens is a person under age 65 who will get the flu this year is 0.0216 (or 2.16%).

(d) To repeat parts (a) and (b) for a community that has 52% senior citizens, we use the same method, but with different probabilities.

For part (a):
P(Senior Citizen) = 52% = 0.52
P(Flu | Senior Citizen) = 10% = 0.10

P(Senior Citizen and Flu) = P(Senior Citizen) * P(Flu | Senior Citizen)
= 0.52 * 0.10
= 0.052

Therefore, the probability that a person selected at random from a community with 52% senior citizens is a senior citizen who will get the flu this season is 0.052 (or 5.2%).

For part (b):
P(Under 65) = 1 - P(Senior Citizen)
= 1 - 0.52
= 0.48

P(Under 65 and Flu) = P(Under 65) * P(Flu | Under 65)
= 0.48 * 0.27
= 0.1296

Therefore, the probability that a person selected at random from a community with 52% senior citizens is a person under age 65 who will get the flu this year is 0.1296 (or 12.96%).