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Answer:
a) There is a 15.3% probability that a randomly selected person in this country is 65 or older.
b) Given that a person in this country is uninsured, there is a 2.2% probability that the person is 65 or older.
Step-by-step explanation:
We have these following percentages:
5.3% of those under the age of 18, 12.6% of those ages 18–64, and 1.3% of those 65 and older do not have health insurance.
22.6% of people in the county are under age 18, and 62.1% are ages 18–64.
(a) What is the probability that a randomly selected person in this country is 65 or older?
22.6% are under 18
62.10% are 18-64
The rest are above 65
So
100% - (22.6% + 62.10%) = 15.3%
There is a 15.3% probability that a randomly selected person in this country is 65 or older.
b) Given that a person in this country is uninsured, what is the probability that the person is 65 or older?
This can be formulated as the following problem:
What is the probability of B happening, knowing that A has happened.
It can be calculated by the following formula
[tex]P = \frac{P(B).P(A/B)}{P(A)}[/tex]
Where P(B) is the probability of B happening, P(A/B) is the probability of A happening knowing that B happened and P(A) is the probability of A happening.
So, what is the probability that a person is 65 and older, given that the person is uninsured.
P(B) is the probability that a person is 65 and older. From a), we have that [tex]P(B) = 0.153[/tex]
P(A/B) is the probability is uninsured, given that that person is 65 and older. So [tex]P(A/B) = 0.013[/tex]
P(A) is the probability that a person is uninsured. That is the sum of 5.3% of 22.6%, 12.6% of 62.1% and 1.3% of 15.3%. So:
[tex]P(A) = 0.053*(0.226) + 0.126*(0.621) + 0.013*(0.153) = 0.0922[/tex]
So
[tex]P = \frac{P(B).P(A/B)}{P(A)} = \frac{0.153*0.013}{0.0922} = 0.022[/tex]
Given that a person in this country is uninsured, there is a 2.2% probability that the person is 65 or older.
Probability of some event is measure of its chance of occurrence. The needed probabilities are:
- The probability that a randomly selected person in this country is 65 or older is: P(C) = 1-P(A)-P(B) = 1-0.847 = 0.153 = 15.3%
- Probability that the person is 65 or older, given that the person is uninsured is: P(C|E) = 0.0053 = 0.53%
How to convert percent to probability?
Percent counts the number compared to 100 whereas probability counts it compare to 1.
So, if we have a%, that means for each 100, there are 'a' parts. If we divide each of them with 100, we get:
For each 1, there are a/100 parts.
Thus, 50% = 50/100 = 0.50 (in probability)
How to calculate the probability of an event?
Suppose that there are finite elementary events in the sample space of the considered experiment, and all are equally likely.
Then, suppose we want to find the probability of an event E.
Then, its probability is given as
[tex]P(E) = \dfrac{\text{Number of favorable cases}}{\text{Number of total cases}}[/tex]
Where favorable cases are those elementary events who belong to E, and total cases are the size of the sample space.
What is the chain rule in probability for two events?
For two events A and B:
The chain rule states that the probability that A and B both occur is given by:
P(A ∩ B) = P(A)P(B|A) = P(B)P( A|B)
P(A|B) means B was given already occurred and we are now calculating probability of A's occurrence, assuming B already occurred.
For the given case, let we take:
E = Event of choosing Uninsured person
A = Event of choosing person with age < 18
B = Event of choosing person with age 18 to 64
C = Event of choosing person with age > 64
Then we have, by given data:
- P(E|A) = 5.3% = 0.053
- P(E|B) = 12.6% = 0.126
- P(E|C) = 1.3% = 0.013
- P(A) = 22.6% = 0.226
- P(B) = 62.1% = 0.621
a) The probability that a randomly selected person in this country is 65 or older is: P(C) = 1-P(A)-P(B) = 1-0.847 = 0.153 = 15.3%
Probability that the person is 65 or older, given that the person is uninsured is: P(C|E) = ?
Now as 5.3% of 22.6% are uninsured, which is 5.3 times 22.6/100 ≈ 1.19%
As 12.6% of 62.1% are uninsured, which is 12.6 times 62.1/100 ≈ 7.82%
And as 1.3% of 15.3% are uninsured, which is 1.3 times 15.3/100 ≈ 0.20
Thus, total uninsured percent = (1.19 + 7.82 + 0.2)% = 9.21%=P(E) = 0.0921
Thus,
P(C ∩ E) = P(E|C))P(C) = P(C|E)P(E)
P(C|E) = P(E|C)P(C)/P(E) = (0.0321 × 0.0153)/0.0921 = 0.0053 approx
Thus, b) Probability that the person is 65 or older, given that the person is uninsured is: P(C|E) = 0.0053 = 0.53%
Thus,
The needed probabilities are:
- The probability that a randomly selected person in this country is 65 or older is: P(C) = 1-P(A)-P(B) = 1-0.847 = 0.153 = 15.3%
- Probability that the person is 65 or older, given that the person is uninsured is: P(C|E) = 0.0053 = 0.53%
Learn more about chain rule of probability here:
https://brainly.com/question/13225049
