Based on data obtained from the National Institute of Dental Research, it has been determined that 44% of 12-year-olds have never had a cavity, 35% of 13-year-olds have never had a cavity, and 25% of 14-year-olds have never had a cavity. If a child is selected at random from a group of 20 junior high school students comprising four 12-year-olds, six 13-year-olds, and ten 14-year-olds and this child does not have a cavity, what is the probability that this child is 14 years old?
A) 0.67
B) 0.04
C) 0.39
D) 0.3

Question

Quizplus · Accepted Answer

We can use Bayes' theorem to solve this problem. Let A be the event that the child is 14 years old, and let B be the event that the child does not have a cavity. We want to find P(A|B), the probability that the child is 14 years old given that they do not have a cavity.

From the given data, we know that:
P(B|A) = 0.25 (the probability of not having a cavity given that the child is 14)
P(B|not A) = 0.35*6/20 + 0.44*4/20 = 0.34 (the probability of not having a cavity given that the child is not 14; we have to weight the probabilities by the number of 13-year-olds and 12-year-olds in the group)

We also know that P(A) = 10/20 = 0.5 (the probability of the child being 14 years old).

Now we can use Bayes' theorem:

P(A|B) = P(B|A)P(A)/[P(B|A)P(A) + P(B|not A)P(not A)]
= 0.25*0.5/[0.25*0.5 + 0.34*0.5]
= 0.39

Therefore, the probability that the child is 14 years old given that they do not have a cavity is 0.39.

Based on Data Obtained from the National Institute of Dental