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Chapter 3: Problem 44

State the additive rule of probability for any two events.

Short Answer

Expert verified
The additive rule of probability is \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).

Step by step solution

01

Introduction to the Additive Rule of Probability

The additive rule of probability is a principle in probability theory which is used to find the probability of the occurrence of at least one of two events. This rule is especially useful when computing probabilities for overlapping events.
02

Define Two Events A and B

Consider two events, A and B, within a sample space. We are interested in finding the probability that either event A occurs or event B occurs, or both events occur.
03

State the Additive Rule Formula

The additive rule of probability can be mathematically expressed as: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]Here, \( P(A \cup B) \) is the probability that either event A or event B (or both) will occur, \( P(A) \) is the probability of event A, \( P(B) \) is the probability of event B, and \( P(A \cap B) \) is the probability that both events A and B occur simultaneously.
04

Explanation of Why We Subtract the Intersection

The term \( P(A \cap B) \) is subtracted because when we add \( P(A) \) and \( P(B) \), the probability of both events occurring together is counted twice. To correct for this overlap, we subtract \( P(A \cap B) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory
Probability Theory helps in understanding and predicting the likelihood of various events. It is a fundamental scientific discipline that evaluates how often we can expect a given event to occur. In simple terms, probability measures how likely an event is to happen. The probability value always ranges between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event.

Through probability theory, we can model complex systems and phenomena by calculating probabilities, which are vital for decision-making in statistics, finance, science, and many more fields.
Sample Space
The sample space is an essential concept in probability theory. It refers to the set of all possible outcomes of a particular experiment or random trial. For example, when you flip a coin, the sample space is {Head, Tail} because there are two possible outcomes.

Understanding the sample space is critical because it lays the groundwork for calculating probabilities. Each outcome in the sample space has an associated probability, and the sum of all probabilities in the space equals one. This ensures that we account for all possible scenarios when evaluating the likelihood of events.
Overlapping Events
Overlapping events occur when two events share some outcomes in common. In probability terms, these are events that can happen simultaneously. For instance, in a deck of cards, drawing a red card and drawing a face card are overlapping events because some face cards are red.

When dealing with overlapping events, it's essential to consider the shared outcomes to avoid counting them twice. This is where the additive rule of probability becomes handy, as it helps correctly calculate the probability of either event occurring.
Intersection of Events
The intersection of events refers to outcomes that are common to two or more events. Mathematically, it is denoted as \( A \cap B \) for events A and B, depicting the set of outcomes that both events share. This concept is crucial because it helps to understand how events relate to one another and how often they occur together.

In probability, calculating the intersection is essential to avoid double-counting probabilities in overlapping events scenarios. For example, in the additive rule, subtracting the probability of the intersection, \( P(A \cap B) \), ensures we get the accurate probability of event A or event B happening.

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Most popular questions from this chapter

Dominant versus recessive traits. A gene is composed of two alleles. An allele can be either dominant or recessive. Suppose a husband and a wife, who are both carriers of the sickle cell anemia allele but do not have the disease, decide to have a child. Because both parents are carriers of the disease, each has one dominant normal cell allele (S) and one recessive sickle cell allele \((s)\). Therefore, the genotype of each parent is \(S s\). Each parent contributes one allele to his or her offspring with each allele being equally likely. a. List the possible genotypes of their offspring. b. What is the probability that the offspring will have sickle cell anemia? In other words, what is the probability that the offspring will have genotype \(s s ?\) Interpret this probability. c. What is the probability that the offspring will not have sickle cell anemia but will be a carrier (one normal cell allele and one sickle cell allele)? Interpret this probability.

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