Question

A study of drug offenders who have been treated for drug abuse suggests that the chance of conviction within a 2 -year period after treatment may depend on the offender's education. The proportions of the total number of cases that fall into four education/conviction categories are shown in the following table. Suppose a single offender is selected from the treatment program. Here are the events of interest: \(A:\) The offender has 10 or more years of education B: The offender is convicted within 2 years after completion of treatment Find the appropriate probabilities for these events: a. \(A\) b. \(B\) c. \(A \cap B\) d. \(A \cup B\) e. \(A^{c}\) f. \(A\) given that \(B\) has occurred g. \(B\) given that \(A\) has occurred

Short answer

Answer: The formula to calculate conditional probability is P(A|B) = P(A ∩ B) / P(B).

Step-by-step solution

Understanding the given information

The table provided shows the proportions of four categories, classified by the offender's education and conviction status. We can use these proportions to calculate the probabilities of events A (the offender has 10 or more years of education) and B (the offender is convicted within 2 years after completion of treatment).

Calculating the probability of A

To find the probability of A (the offender has 10 or more years of education), we will sum up the proportions of cases with 10 or more years of education, regardless of whether the offender is convicted or not. If we denote the proportions as \(p_{1}, p_{2}, p_{3}, p_{4}\) according to their categories, then \(P(A) = p_{1} + p_{2}\).

Calculating the probability of B

To find the probability of B (the offender is convicted within 2 years after completion of treatment), we sum up the proportions of cases where the offender is convicted, regardless of their education level. So, \(P(B) = p_{2} + p_{4}\).

Calculating the probability of A ∩ B

The probability of A ∩ B means the probability of an offender having both 10 or more years of education and being convicted within 2 years after treatment. This corresponds to category 2 in the table, so \(P(A \cap B) = p_{2}\).

Calculating the probability of A ∪ B

To find the probability of A ∪ B (either A or B or both), we can use the formula for the union of probabilities: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).

Calculating the probability of \(A^{c}\)

The probability of \(A^{c}\), the complement of A, represents the probability of an offender having less than 10 years of education. We can find this by subtracting the probability of A from 1: \(P(A^{c}) = 1 - P(A)\).

Calculating the probability of A given B

To find the conditional probability of A given B (\(P(A|B)\)), meaning the probability of an offender having 10 or more years of education given that they were convicted within 2 years after treatment, we use the formula for conditional probability: \(P(A|B) = \frac{P(A \cap B)}{P(B)}\).

Calculating the probability of B given A

Lastly, to find the conditional probability of B given A (\(P(B|A)\)), meaning the probability of an offender being convicted within 2 years after treatment given that they have 10 or more years of education, we use the formula for conditional probability: \(P(B|A) = \frac{P(A \cap B)}{P(A)}\).

Key concepts

Conditional Probability

Conditional probability represents the likelihood of one event happening, given that another event has already occurred. It's written as \(P(A|B)\) to express "the probability of event A given that B has occurred."

To calculate conditional probability, you use the equation: \[P(A|B) = \frac{P(A \cap B)}{P(B)}\] Here, \(P(A \cap B)\) is the probability of both events A and B happening at the same time. And \(P(B)\) is the probability of B occurring. By using the conditional probability formula, you can refine the likelihood of an event when new, related information is available.

This concept comes in handy in situations like predicting outcomes based on previous conditions or narrowing down possibilities in decision-making.

Event Probability

Event probability reflects the chance of a single outcome happening in a particular experiment or situation. Let's think of it as the basic building block of probability.

To determine event probability, you calculate: \[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\] In situations like the one presented with offenders, if you want to know the probability that the offender has 10 or more years of education, you sum up the proportions from relevant categories.

Knowing event probabilities helps in predicting certain outcomes and preparing for potential scenarios.

Union and Intersection of Events

Union and intersection describe how two events interact. The **union** of events, expressed as \(A \cup B\), involves all outcomes that belong to either event A or event B, or to both. The formula for union is:\[ P(A \cup B) = P(A) + P(B) - P(A \cap B)\] Here, the minus part \(P(A \cap B)\) ensures that any overlap (where both events happen) is not counted twice.

The **intersection** of events, expressed as \(A \cap B\), captures only the outcomes that events A and B share, meaning both events happen together.

Understanding these concepts allows you to evaluate how events relate and influence each other. This is crucial when analyzing situations with multiple criteria or factors.

Complementary Events

Complementary events cover all the probabilities that aren't included in the desired event. If you have event A, the complement is denoted as \(A^c\) and describes everything outside of A. The two, A and \(A^c\), collectively cover all possible outcomes.

The formula is simply: \[ P(A^c) = 1 - P(A)\] Using \( (1-P(A)) \), you effectively account for the uncertainty not captured by A, ensuring complete coverage of the scenario.

Complementary events are useful in many real-world applications, such as confidence intervals and risk assessments, because they provide insight into the remainder of potential outcomes that are not originally targeted.