91Ó°ÊÓ

Suppose that \(20 \%\) of all teenage drivers in a certain county received a citation for a moving violation within the past year. Assume in addition that \(80 \%\) of those receiving such a citation attended traffic school so that the citation would not appear on their permanent driving record. Consider the chance experiment that consists of randomly selecting a teenage driver from this county. a. One of the percentages given in the problem specifies an unconditional probability, and the other percentage specifies a conditional probability. Which one is the conditional probability, and how can you tell? b. Suppose that two events \(E\) and \(F\) are defined as follows: \(E=\) selected driver attended traffic school \(F=\) selected driver received such a citation Use probability notation to translate the given information into two probability statements of the form \(P(\underline{C})=\) probability value.

Step by step solution

01

Unconditional probability

The unconditional probability is the probability that does not depend on any other event. In this case, the unconditional probability is the percentage of teenage drivers receiving a citation for a moving violation within the past year, which is \(20\%\).

02

Conditional probability

The conditional probability is the probability that depends on another event. In this case, the conditional probability is the percentage of those receiving citations and attending traffic school so that citation would not appear on their permanent driving record, which is \(80\%\). We can tell this is the conditional probability because it only applies to the group of teenage drivers who received a citation. #b. Translating information into probability statements#

03

Event E probability

Event E states that the selected driver attended traffic school. We need to find the probability that a randomly selected teenage driver attended traffic school. We know that \(80\%\) of those who received citations attended traffic school, and \(20\%\) of all teenage drivers received citations. Therefore, the probability of a teenage driver attending traffic school can be found using the formula \(P(E|F) = \frac{P(E \cap F)}{P(F)}\). We want to find \(P(E \cap F)\), the probability of both E and F happening. We have: \(P(E \cap F)= P(E|F) \cdot P(F) = 0.80 \cdot 0.20 = 0.16\)

04

Event F probability

Event F states that the selected driver received a citation. The probability that a randomly selected teenage driver received a citation is given as \(20\%\). So, we have: \(P(F) = 0.20\) The two probability statements are: - \(P(E \cap F)= 0.16\) - \(P(F)=0.20\)

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Key Concepts

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

Conditional Probability

Conditional probability is a way of expressing how likely an event is to happen given that another event has already occurred. It's an essential concept in probability theory that helps us analyze events that are interdependent.

In our exercise, conditional probability is illustrated by looking at teenage drivers who have received a citation. Among these, 80% attended traffic school. This is a conditional probability because it specifically concerns only those drivers who have already received a citation. The condition here is the prior event of receiving a citation.

To calculate the conditional probability, we use the formula \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), where \( A \) and \( B \) are two events, and \( A|B \) represents the probability of \( A \) occurring given \( B \) has occurred. This formula helps in understanding the probability of an event under a specific condition.

Unconditional Probability

Unconditional probability, sometimes referred to as marginal probability, assesses the likelihood of an event without considering any other events. It provides a baseline probability for an event occurring.

In the given exercise, the unconditional probability is the percentage of teenage drivers who received a moving violation citation over the past year. This value is given as 20%.

This probability is considered unconditional as it isn't influenced or conditioned by other events. It represents a straightforward, singular measure of likelihood based only on one criterion, which is receiving a citation in this context. Recognizing unconditional probabilities is crucial for building a comprehensive understanding of more complex probability scenarios, including conditional ones.

Probability Notation

Probability notation is a way to express the likelihood of certain outcomes in a concise mathematical format. This allows for the succinct communication of probability concepts and the handling of complex probability calculations.

In our exercise, we express probabilities using symbols and equations. For instance, the probability that a selected driver received a citation is denoted as \( P(F) = 0.20 \).

Furthermore, the joint probability that a driver both attended traffic school and received a citation uses the intersection notation: \( P(E \cap F) = 0.16 \). Probabilities are often followed by a letter or string in parentheses, identifying the event. Notation helps differentiate between types of probabilities. For example, \( P(E|F) \) signifies a conditional probability, where event \( E \) is dependent on event \( F \) having occurred.

Traffic School Attendance

Understanding the concept of traffic school attendance is important in the exercise's context as it is tied to the consequences of citations for teenage drivers. Traffic school attendance often affords young drivers the opportunity to mitigate the long-term impact of their violations.

In the scenario given, 80% of cited drivers attend traffic school. This reflects the conditional nature of attendance—those attending school have already been issued citations. Traffic school attendance serves as a corrective measure that influences the driver's record, emphasizing not just the probability but also the action taken by the drivers.

This concept links directly to real-world implications and is more appreciated when explored through the lens of conditional probability. The likelihood of having attended traffic school is calculated using the probabilities involved with receiving citations.

Event Probability

Event probability refers to the likelihood that a specific event will occur within a given chance experiment. It is a fundamental part of probability theory.

For example, in the exercise, two events are identified: \( E \), the event of attending traffic school, and \( F \), the event of receiving a citation. Understanding the probability of each of these events gives insight into the behavior of teenage drivers in the context.

The exercise provides these probabilities directly or calculates them through simple probability operations. Knowing how to determine event probabilities assists in forming the probability landscape for more comprehensive chance experiments. It's crucial for grasping more complex concepts such as joint and conditional probabilities in applied and theoretical probabilities alike.