Calculating Probability P(A Or B) With Examples And Explanations

We are given the following probabilities:

• P(A) = 0.50
• P(B) = 0.40
• P(A and B) = 0.15

We need to find P(A or B).

Solution

The formula for the probability of A or B is:

P(A or B) = P(A) + P(B) - P(A and B)

This formula is crucial in probability theory and helps us calculate the likelihood of either event A or event B occurring, or both. It's essential for understanding how events interact and overlap. The formula is derived from the principle of inclusion-exclusion, which ensures that we don't double-count the probability of the intersection of the two events. This principle is a cornerstone of probability calculations and is widely applicable in various fields, including statistics, data analysis, and risk assessment. A deep understanding of this formula allows for accurate predictions and informed decision-making in scenarios involving uncertainty. By applying this formula correctly, we can effectively analyze and interpret probabilistic events in a wide range of contexts, from simple coin flips to complex real-world situations. In many practical scenarios, events are not mutually exclusive, meaning they can occur simultaneously. In such cases, the probability of either event occurring is not simply the sum of their individual probabilities. We must account for the overlap, which is the probability of both events happening together. This overlap is represented by P(A and B). The inclusion-exclusion principle ensures that we subtract this overlap to avoid double-counting it. This nuanced approach is vital for accurate probability calculations, especially in situations where events are interdependent. For instance, consider the probability of drawing a heart or a king from a deck of cards. There is one card that is both a heart and a king, so we need to subtract the probability of drawing this card to get the correct answer. The correct application of this formula demonstrates a strong grasp of probability concepts and their real-world implications. Using this formula allows for a more precise understanding of the likelihood of compound events, where multiple outcomes are possible. This is essential in fields such as insurance, where risks must be accurately assessed, and in scientific research, where experimental results must be interpreted. By understanding the principles behind the formula, we can make better predictions and decisions in situations involving uncertainty, making it a powerful tool in our analytical toolkit.

Substituting the given values:

P(A or B) = 0.50 + 0.40 - 0.15

P(A or B) = 0.90 - 0.15

P(A or B) = 0.75

Therefore, the probability of A or B is 0.75.

Answer

The correct answer is C. 0.75

Probability P(A or B), denoted as P(A ∪ B), represents the likelihood that either event A, event B, or both events occur. This concept is fundamental in probability theory and statistics, allowing us to analyze the chances of multiple events happening. Understanding how to calculate P(A or B) is crucial in various applications, from predicting weather patterns to assessing risks in financial markets. The calculation involves considering the individual probabilities of events A and B, as well as the probability of their intersection, P(A and B). This ensures that we accurately account for the scenarios where both events occur simultaneously. In many real-world situations, events are not mutually exclusive, meaning they can happen at the same time. Therefore, we need to use the formula P(A or B) = P(A) + P(B) - P(A and B) to avoid double-counting the overlapping outcomes. This formula is derived from the principle of inclusion-exclusion, a cornerstone of combinatorial mathematics and probability theory. The principle of inclusion-exclusion ensures that we correctly account for all possible outcomes without overcounting. It is particularly useful when dealing with complex scenarios involving multiple events and their interactions. The ability to accurately calculate P(A or B) is essential for making informed decisions in various fields, including medicine, engineering, and social sciences. For example, in medical research, it helps in determining the probability of a patient experiencing a particular symptom or side effect. In engineering, it aids in assessing the reliability of systems by calculating the probability of component failures. In social sciences, it can be used to analyze survey data and predict outcomes based on different factors. Understanding and applying the concept of P(A or B) is a vital skill for anyone working with data and probabilities.

To further illustrate, let's consider a few examples:

Example 1: Rolling a Die

Suppose we roll a six-sided die. Let event A be rolling an even number (2, 4, or 6), and event B be rolling a number greater than 3 (4, 5, or 6).

• P(A) = 3/6 = 0.5
• P(B) = 3/6 = 0.5
• P(A and B) = P(rolling a 4 or 6) = 2/6 = 1/3

P(A or B) = P(A) + P(B) - P(A and B) = 0.5 + 0.5 - (1/3) = 1 - (1/3) = 2/3

This example demonstrates how the formula accounts for the overlap between the two events. Without subtracting P(A and B), we would be double-counting the outcomes 4 and 6.

Example 2: Drawing a Card

Consider drawing a card from a standard 52-card deck. Let event A be drawing a heart, and event B be drawing a king.

• P(A) = 13/52 = 1/4
• P(B) = 4/52 = 1/13
• P(A and B) = P(drawing the king of hearts) = 1/52

P(A or B) = P(A) + P(B) - P(A and B) = (1/4) + (1/13) - (1/52) = (13/52) + (4/52) - (1/52) = 16/52 = 4/13

In this case, there is one card that is both a heart and a king (the king of hearts), so we need to subtract its probability to get the correct answer.

Example 3: Students and Subjects

In a class, 60% of students like mathematics, and 40% like science. If 20% like both subjects, what is the probability that a randomly selected student likes either mathematics or science?

• P(Mathematics) = 0.60
• P(Science) = 0.40
• P(Mathematics and Science) = 0.20

P(Mathematics or Science) = P(Mathematics) + P(Science) - P(Mathematics and Science) = 0.60 + 0.40 - 0.20 = 0.80

Therefore, 80% of the students like either mathematics or science.

Understanding the concept of P(A or B) is critical in probability theory because it allows us to calculate the likelihood of at least one of several events occurring. This is a fundamental concept that extends beyond simple scenarios and is essential for more complex probabilistic analyses. The ability to determine the probability of the union of events is a cornerstone of statistical inference, risk assessment, and decision-making under uncertainty. By mastering the formula and its applications, one can gain deeper insights into probabilistic systems and make more informed predictions. The formula for P(A or B), which is P(A or B) = P(A) + P(B) - P(A and B), is derived from the principle of inclusion-exclusion. This principle ensures that we do not double-count the outcomes that are common to both events A and B. The intersection of events, represented by P(A and B), plays a crucial role in accurately calculating the probability of the union. Failing to account for this intersection can lead to an overestimation of the probability of either event occurring. In many real-world applications, events are not mutually exclusive; that is, they can occur simultaneously. For instance, consider the probability of a company's stock price increasing or a new product launch being successful. These events are not mutually exclusive, as both can happen in the same time frame. Therefore, understanding how to calculate P(A or B) in non-mutually exclusive scenarios is crucial for making realistic predictions and informed decisions. The concept of P(A or B) is also essential in hypothesis testing, where we often need to determine the probability of observing a certain outcome or a more extreme outcome, given that a particular hypothesis is true. This is fundamental in scientific research and statistical analysis. The correct application of P(A or B) allows researchers to make accurate conclusions about their hypotheses. Furthermore, in risk management, understanding P(A or B) helps in assessing the likelihood of potential risks and developing strategies to mitigate them. For example, in financial risk management, calculating the probability of market downturns or credit defaults is essential for portfolio optimization and risk control. In engineering, P(A or B) is used to assess the reliability of systems and the likelihood of failures, enabling the design of more robust and safe systems. Overall, the ability to understand and apply the concept of P(A or B) is a cornerstone of probability theory and has wide-ranging implications across various disciplines. By mastering this concept, individuals can better analyze data, make predictions, and make informed decisions in uncertain environments.

In conclusion, understanding the formula and application of P(A or B) is essential for solving probability problems and gaining a deeper understanding of probabilistic events. By correctly applying the formula and considering the intersection of events, we can accurately calculate the likelihood of either event A or event B occurring.