Problem 38 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Based on data obtained from the National Institute of Dental Research, it has been determined that \(42 \%\) of 12 -yr-olds have never had a cavity, \(34 \%\) of 13 -yr-olds have never had a cavity, and \(28 \%\) of 14 -yrolds have never had a cavity. Suppose a child is selected at random from a group of 24 junior high school students that includes six 12 -yr-olds, eight 13 -yr-olds, and ten 14 -yrolds. If this child does not have a cavity, what is the probability that this child is 14 yrs old?

Step-by-Step Solution

Answer

The probability that the child is 14 years old, given that they do not have a cavity, is approximately \(0.306\).

Step 1: Calculate the probabilities of selecting each age group

First, we calculate the probabilities of selecting a 12-yr-old, 13-yr-old, and 14-yr-old child. There are 24 children in total, with six 12-yr-olds, eight 13-yr-olds, and ten 14-yr-olds. Using these values, Probability of selecting a 12-yr-old: \(P(A_1) = \frac{6}{24}\) Probability of selecting a 13-yr-old: \(P(A_2) = \frac{8}{24}\) Probability of selecting a 14-yr-old: \(P(A_3) = \frac{10}{24}\)

Step 2: Calculate the probabilities of not having a cavity for each age group

We are given the probabilities of not having a cavity for each age group: For 12-yr-olds: \(P(B|A_1) = 0.42\) For 13-yr-olds: \(P(B|A_2) = 0.34\) For 14-yr-olds: \(P(B|A_3) = 0.28\)

Step 3: Calculate the probability of selecting a child without a cavity

For this, we can use the total probability formula, which states that \(P(B)= P(A_1)P(B|A_1) + P(A_2)P(B|A_2) + P(A_3)P(B|A_3)\) So, \(P(B)= \frac{6}{24}(0.42) + \frac{8}{24}(0.34) + \frac{10}{24}(0.28)\)

Step 4: Apply Bayes' theorem to calculate the probability of a 14-year-old child given they don't have a cavity

Using Bayes' theorem, the probability of selecting a 14-year-old given that the child doesn't have a cavity is: \(P(A_3|B) = \frac{P(A_3) P(B|A_3)}{P(B)}\) Plug in the values that we obtained in steps 1, 2, and 3: \(P(A_3|B) = \frac{\frac{10}{24}(0.28)}{\frac{6}{24}(0.42) + \frac{8}{24}(0.34) + \frac{10}{24}(0.28)}\) Finally, after computing the above expression, you'll get the probability that the child is 14 years old: \(P(A_3|B) \approx 0.306\)

Key Concepts

Understanding Conditional Probability

Conditional probability is a fundamental concept in probability and statistics that refers to the probability of an event occurring, given that another event has already occurred. This concept goes beyond the likelihood of events happening in isolation and addresses how the occurrence of one event can impact the occurrence of another.

In our example, we consider the probability of a child being 14 years old given that they do not have a cavity, expressed as \(P(A_3|B)\). Here \(A_3\) is the event where the child is 14 years old, and \(B\) is the event of a child not having a cavity. The vertical bar '|' stands for 'given that'. Calculating conditional probabilities involves dividing the probability of both events occurring simultaneously by the probability of the known condition.

Key Takeaways on Conditional Probability:

• Conditional probability quantifies the chance of one event under the assumption that another event has occurred.
• It's notated as \(P(A|B)\), representing the probability of event \(A\) occurring given that \(B\) has occurred.
• Understanding this concept is crucial for more complex probability theories, such as Bayes' theorem.

Applying the Total Probability Formula

The total probability formula is invaluable for calculating the likelihood of an event based on several distinct outcomes or scenarios. It essentially breaks down a complex probability into simpler parts that are easier to handle. The formula sums up the probabilities of the event occurring due to each individual outcome.

In the exercise, the total probability formula is used to determine the probability of randomly selecting a child without a cavity from the entire junior high school group. The solution involves summing up the probabilities of a child not having a cavity across all age groups, taking into account both the probability of selecting a child from each age group and the respective probabilities of a child in each age group not having a cavity.

Remember This:

• Total probability formula combines individual probabilities from related scenarios to find the overall probability of an event.
• It's particularly useful when dealing with multiple conditions that can lead to the same outcome.
• In our example, the formula helps us to ascertain the likelihood of selecting a cavity-free child, irrespective of the age group.

Demystifying Bayes' Theorem

Bayes' theorem is a powerful concept in probability that allows us to update our previous beliefs or probabilities based on new evidence or information. It ties together conditional probabilities in a way that lets us reverse them: we can find out the probability of the cause given the effect. This theorem is named after Reverend Thomas Bayes, who formulated it in the 18th century.

The theorem is central to understanding our example problem, where we need to find the probability that a child is 14 years old given they don't have a cavity. By applying Bayes' theorem, we can use the known probabilities to calculate this revised probability.

Bayes' Theorem Essentials:

• Allows for the computation of 'reverse' conditional probabilities.
• Essential for many applications in statistics, science, and engineering.
• In our context, it dissects the general likelihood of a child not having a cavity to find the specific probability for 14-year-olds.

Applying the theorem to the problem, we find that the probability of a child being 14 years old without a cavity is approximately 30.6%, showcasing how Bayes' theorem can provide insight into specific conditional probabilities.