Problem 51 - Algebra and Trigonometry

Question

According to the Insurance Information Institute, in 2016 there was a \(13.3 \%\) probability that an automobile theft in the United States would be cleared by arrests. If an automobile theft case from 2016 is randomly selected, what is the probability that it was not cleared by an arrest?

Step-by-Step Solution

Answer

The probability that the automobile theft case was not cleared by arrest is 0.867 or 86.7%.

Step 1 - Understand the Problem

An event has a given probability of occurring. The task is to find the complement probability that the event does not occur. Here, the event is an automobile theft case being cleared by arrest, with a probability of 13.3%.

Step 2 - Identify the Given Probability

The probability that an automobile theft case in 2016 was cleared by arrest is 13.3%. This can be written as a decimal: \[P(\text{cleared by arrest}) = 0.133\]

Step 3 - Calculate the Complement Probability

The complement of an event occurring is calculated as 1 minus the probability of the event occurring. So, to find the probability that the case was not cleared by arrest, use the formula: \[P(\text{not cleared by arrest}) = 1 - P(\text{cleared by arrest})\].Substitute the given probability: \[P(\text{not cleared by arrest}) = 1 - 0.133 = 0.867\]

Key Concepts

Complement Probability

In probability theory, the complement of an event is all the possible outcomes that are not part of the event you are considering. For example, if we have an event A with probability, \(P(A)\), the complement of event A is denoted by \(P(A^c) \) and represents the probability that event A does not occur. To find this complement probability, use the formula \(P(A^c) = 1 - P(A) \). This is because the total probability of all possible outcomes must sum to 1.

In our exercise, the event is an automobile theft case being cleared by arrest with a probability of 13.3%. The complement probability, or the likelihood that an automobile theft case was not cleared by arrest, is calculated as follows: \[ P(\text{not cleared by arrest}) = 1 - P(\text{cleared by arrest}) = 1 - 0.133 = 0.867 \].

So, you would have an 86.7% chance that a randomly chosen car theft case was not solved by an arrest in 2016.

Probability Calculation

Calculating probabilities can initially seem intimidating, but it's often simpler than it looks. The first step is always converting percentages to decimals by dividing by 100. For instance, in our example, the probability of a theft being cleared by arrest is given as 13.3%, which converts to 0.133 in decimal form.

A critical concept to grasp is the use of the complement rule for calculating the probability of the event not occurring. Given the probability of an event A, \(P(A)\), the probability of the event not occurring is found by subtracting \(P(A)\) from 1. The formula to keep in mind is: \[P(A^c) = 1 - P(A)\].

If you take the example from our exercise:

• Given probability, \(P(\text{cleared by arrest}) = 0.133 \)
• Complement probability, \(P(\text{not cleared by arrest}) = 1 - 0.133 = 0.867 \)

This method of calculation helps you determine both the occurrence and non-occurrence probabilities efficiently.

Event Occurrence

An event in the context of probability refers to any specific set of outcomes of a random phenomenon. For example, in our exercise, the event of interest is 'an automobile theft being cleared by an arrest'. Each event has an associated probability that quantifies how likely it is to happen.

Event occurrence is represented by \(P(E) \), meaning the probability of event E occurring. Probabilities always range from 0 to 1, with 0 meaning the event will not occur and 1 meaning the event will definitely occur.

Understanding how to assess and calculate the probability of an event's occurrence is essential in many fields, including statistics, insurance, finance, and everyday decision-making. In our given exercise:

• Event: An automobile theft being cleared by an arrest in 2016.
• Probability of occurrence, \(P(\text{cleared by arrest}) = 0.133 \)
• Probability of non-occurrence (complement), \(P(\text{not cleared by arrest}) = 0.867 \)

This knowledge helps in understanding the broader context where probability is applied, helping in making informed decisions and predictions.