Question

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 \(P(\ldots)=\) probability value.

Short answer

The conditional probability is the chance a driver attended traffic school given they received a citation. It is conditional because attending traffic school depends on if they received a citation. The event of receiving a citation itself is an unconditional probability. The probability statements for the events \(E\) and \(F\) are \( P(E|F) = 0.80 \) and \( P(F) = 0.20 \) respectively.

Step-by-step solution

Identify the Conditional Probability

Out of the given percentages, the probability of 'a random teenage driver attending a traffic school given that they received a moving violation citation' is the conditional probability. This is because the event of attending traffic school relies on the event that they received citation. This is expressed as \( P(E|F) = 0.80 \).

Identify the Unconditional Probability

The probability 'a random teenage driver receiving a moving violation citation' is an unconditional probability since it does not depend on any other event. This is stated as \( P(F) = 0.20 \).

Translate events into probability statements

Now, we can translate the given information into probability statements. The events \(E\) and \(F\) can be represented as: \( P(E|F) = 0.80 \) and \( P(F) = 0.20 \). These are the probability statements for the provided events.

Key concepts

Unconditional Probability

Unconditional probability, also known simply as probability, represents the likelihood of an event occurring without considering the influence of any other events. It's a foundation stone in the world of probability and statistics, giving us the starting point to understanding more complex probability concepts.

For instance, in the context of our exercise, when we talk about the percentage of teenage drivers in a county receiving a moving violation, we're referring to an unconditional probability. It is not dependent on or conditioned by any other event. Mathematically, it is represented as \( P(F) = 0.20 \).This number is describing the odds of any randomly chosen teenage driver having received a citation, out of the entire population of drivers. It's important for students to understand that unconditional probabilities will always have values between 0 and 1, where 0 means the event cannot happen and 1 means the event is certain to happen.

Probability Notation

Understanding probability notation is key to communicating complicated probability concepts succinctly and precisely. In the exercise, we are given the probabilities in percentage form, but typically, probabilities are expressed using the 'P' notation.For example, the unconditional probability that we discussed before is denoted as \( P(F) \),which reads as 'the probability of event F occurring.' Similarly, conditional probabilities are denoted with a vertical bar, often read as 'given.' An example from the exercise would be the probability of a teenage driver attending traffic school given they have received a citation, written as \( P(E|F) \),and is equal to 0.80. This notation is crucial for students to learn as it will be used consistently in the field of probability. It allows for a clear, concise way to breakdown and understand probability questions and their respective solutions.

Probability Statements

Probability statements are expressions that concisely convey the likelihood of events occurring. They use probability notation to clarify the relationship between different events, which might be independent or dependent.In our exercise, we turn the given percentages into probability statements, such as \( P(E|F) = 0.80 \),which states that there is an 80% chance that a teenage driver attends traffic school given they've received a citation. This is an example of how probability statements can represent conditional probabilities.Additionally, the statement \( P(F) = 0.20 \),translates to a 20% chance of a randomly selected teenage driver receiving a citation. These statements transform the conceptual understanding of probability into a form that can be easily analyzed and applied, and are an essential tool for anyone studying probability.