Question

Use the information that, for events \(\mathrm{A}\) and \(\mathrm{B}\), we have \(P(A)=0.8, P(B)=0.4\), and \(P(A\) and \(B)=0.25.\) Are events \(\mathrm{A}\) and \(\mathrm{B}\) disjoint?

Short answer

No, events \(A\) and \(B\) are not disjoint.

Step-by-step solution

Check Intersection Probability

The probability of intersection of events A and B, denoted as \(P(A \cap B)\), is given as 0.25 from the exercise. Disjoint events have an intersection probability of 0. So, since the intersection probability here is not 0, we can conclude that events A and B are not disjoint.

Key concepts

Intersection Probability

When studying probability, the term intersection probability is vital. It refers to the likelihood of two events occurring simultaneously. In a more formal language, the intersection of events A and B, denoted as \( P(A \cap B) \), represents the probability that both A and B occur at the same time. If you roll a die, for example, the probability that you roll an even number (event A) and a number greater than 2 (event B) at the same time is an intersection probability.

To assess whether events are disjoint (mutually exclusive), we look at their intersection probability. Disjoint events cannot happen at the same time, which means that if A and B are disjoint, \( P(A \cap B) = 0 \). However, in our case, we have \( P(A \cap B) = 0.25 \), indicating that events A and B are indeed not disjoint, because there is an overlap in outcomes represented by the positive intersection probability.

Probability Theory

At the core of probability theory is the study of uncertainty and randomness. It is a mathematical framework for quantifying and analyzing the likelihood of various outcomes. In this framework, probabilities are assigned to a set of outcomes of a random experiment. It plays a critical role in a wide array of disciplines from gaming to statistics, economics, and even weather forecasting.

Key principles within probability theory are concepts such as events, outcomes, and sample spaces. Probability values range from 0 (an event that will never occur) to 1 (an event that is certain to occur). For instance, in our original exercise, the fact that \( P(A) = 0.8 \) suggests that event A has a high likelihood of occurring, while \( P(B) = 0.4 \) indicates that event B is less likely in comparison. Probability theory helps us understand and predict how likely multiple events will interact, such as when evaluating the intersection of events A and B.

Statistics Principles

Moving into the realm of statistics principles, we transition from theory to application. Statistics uses probability theory as a foundation to analyze data and draw conclusions about larger populations based on samples. It helps in making decisions when faced with uncertainty.

When applying statistics principles to probabilistic events, the concept of independence and disjointness is crucial. Two events are statistically independent if the occurrence of one does not affect the occurrence of the other, which in turn plays a role in calculations such as the multiplication rule for independent events. In contrast, as mentioned earlier, events are disjoint if they cannot occur simultaneously.

Understanding these principles is key in fields such as data science and economics, where making predictions based on probability models can have significant implications. As such, the intersection probability in our exercise is more than just a mathematical concept; it is a practical tool that helps statisticians gauge the relationship between events, crucial for accurate data interpretation.