Use Bayes' theorem to find the indicated probability.
-In one town, 4% of 18-29 year olds own a home, as do 25% of 30-50 year olds and 52% of those over 50. According to a recent census taken in the town, 26.2% of adults in the town are 18-29 years old, 36.0% are 30-50
Years old, and 37.8% are over 50. If an adult is selected at random from the town and found to be a
Home-owner, what is the probability that they are aged between 30 and 50?
A) 0.09
B) 0.36
C) 0.25
D) 0.303

Question

Quizplus · Accepted Answer

We are looking for the probability of an individual being a homeowner given they are aged between 30 and 50. Using Bayes' Theorem:

P(30-50 | Homeowner) = P(Homeowner | 30-50) * P(30-50) / P(Homeowner)

We already know the following probabilities:
P(18-29 | Homeowner) = 0.04
P(30-50 | Homeowner) = 0.25
P(over 50 | Homeowner) = 0.52
P(18-29) = 0.262
P(30-50) = 0.36
P(over 50) = 0.378

To find P(Homeowner), we can use the Law of Total Probability:

P(Homeowner) = P(Homeowner | 18-29) * P(18-29) + P(Homeowner | 30-50) * P(30-50) + P(Homeowner | over 50) * P(over 50)
= (0.04)(0.262) + (0.25)(0.36) + (0.52)(0.378)
= 0.31596

Now we can substitute all the probabilities into Bayes' Theorem:

P(30-50 | Homeowner) = (0.25)(0.36) / 0.31596
= 0.303

Therefore, the probability of an individual being aged between 30 and 50 given they are a homeowner is 0.303, which corresponds to answer choice D.

Use Bayes' Theorem to Find the Indicated Probability