Problem 31 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Data compiled by the Department of Justice on the number of people arrested in a certain year for serious crimes (murder, forcible rape, robbery, and so on) revealed that \(89 \%\) were male and \(11 \%\) were female. Of the males, \(30 \%\) were under 18 , whereas \(27 \%\) of the females arrested were under 18 . a. What is the probability that a person arrested for a serious crime in that year was under 18 ? b. If a person arrested for a serious crime in that year is known to be under 18 , what is the probability that the person is female?

Step-by-Step Solution

Answer

The probability that a person arrested for a serious crime in that year was under 18 is approximately \(29.88\% \), and the probability that a person arrested for a serious crime in that year is known to be under 18 is female is approximately \(9.97\% \).

Step 1: Identify the events

Let's identify the relevant events for this problem: - M: A person arrested is male. - F: A person arrested is female. - U: A person arrested is under 18.

Step 2: Calculate the probability of each gender

We are given the probability of a person arrested being male or female. We have: - \(P(M) = 0.89\) - \(P(F) = 0.11\)

Step 3: Calculate the probability of being under 18 given the gender

We are also given the probability of being under 18 given the person's gender: - \(P(U|M) = 0.3\) - \(P(U|F) = 0.27\)

Step 4: Calculate the overall probability of being under 18 using the law of total probability

By the law of total probability, we can find the overall probability of a person arrested being under 18, as: \[P(U) = P(U|M)P(M) + P(U|F)P(F)\] Plugging in the values we calculated earlier: \[P(U) = (0.3)(0.89) + (0.27)(0.11)\] Calculate the probability: \[P(U) \approx 0.2988\] So, the probability that a person arrested for a serious crime in that year was under 18 is approximately \(29.88\% \).

Step 5: Calculate the probability of being female given being under 18 using Bayes' theorem

Now we will find the probability that a person arrested for a serious crime in that year is known to be under 18 is female, using Bayes' theorem: \[P(F|U) = \frac{P(U|F)P(F)}{P(U)}\] Plugging in the values we calculated earlier: \[P(F|U) = \frac{(0.27)(0.11)}{0.2988}\] Calculate the probability: \[P(F|U) \approx 0.0997\] So, the probability that a person arrested for a serious crime in that year is known to be under 18 is female is approximately \(9.97\% \).

Key Concepts

Bayes' Theorem

Bayes' Theorem is a powerful tool in probability that allows us to update our beliefs based on new evidence. It's fundamentally about conditional probability and helps in calculating the probability of an event, given that another event has occurred. This theorem is especially useful when the reverse conditional probability is easier to determine.

In mathematical terms, Bayes' Theorem can be stated as:

• \[ P(A|B) = \frac{P(B|A)P(A)}{P(B)} \]

Where:

• \( P(A|B) \) is the probability of event A occurring given B is true.
• \( P(B|A) \) is the probability of event B occurring given A is true.
• \( P(A) \) and \( P(B) \) are the probabilities of events A and B independently of each other.

The exercise asks to find the probability that the individual arrested is female given they are under 18, noted as \( P(F|U) \). Using Bayes' Theorem, this probability is calculated as:

• \[ P(F|U) = \frac{P(U|F)P(F)}{P(U)} \]

Here, \( P(U|F) \) represents the probability that an individual is under 18 given they are female, \( P(F) \) is the probability of being female, and \( P(U) \) is the total probability of being under 18. This formulation allows us to derive the likelihood of gender given age, utilizing known probabilities.

Law of Total Probability

The Law of Total Probability is a fundamental rule that provides a way to break down complex probability calculations into simpler parts. It is especially useful when assessing the probability of an event when the event space can be partitioned into disjoint subsets, like different categories or groups.

The law states that if \( \{B_1, B_2, \ldots, B_n\} \) are mutually exclusive and collectively exhaustive events, the probability of event A can be calculated as the sum of the conditional probabilities of A given each \( B_i \), weighted by the probability of each \( B_i \):

• \[ P(A) = \sum_{i=1}^{n} P(A|B_i)P(B_i) \]

For our exercise, the law was applied to find the total probability \( P(U) \), which represents the likelihood of someone being under 18:

• \[ P(U) = P(U|M)P(M) + P(U|F)P(F) \]

Here, \( P(U|M) \) and \( P(U|F) \) are the probabilities of being under 18 for males and females, respectively, and \( P(M) \) and \( P(F) \) are the probabilities of being male and female. Summing these weighted conditions helps to cover the entire arrested population under the age of 18.

Conditional Probability

Conditional probability is fundamental in probability theory, representing the probability of an event occurring, given that another event has already occurred. It reflects how the likelihood of events is interconnected or influenced when additional information is available.

Mathematically, if you want to find the probability of event A given event B, it can be expressed as:

• \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

Where:

• \( P(A \cap B) \) is the probability that both events A and B occur.
• \( P(B) \) is the probability of event B occurring.

In the given exercise, conditional probability plays a vital role in interpreting events, such as determining the likelihood of an individual being under the age of 18 given their gender. This correlation between one's age and gender in the context of the dataset reveals how one event (being under 18) is probably influenced by another (being male or female). By considering these conditional setups, we understand how certain conditions impact broader probability calculations, which can be applied in real-world data assessment scenarios.