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According to a study of 100 drivers in metropolitan Washington D.C. whose cars were equipped with cameras with sensors, the distractions and the number of incidents (crashes, near crashes, and situations that require an evasive maneuver after the driver was distracted) caused by these distractions are as follows: $$ \begin{array}{lccccccccc} \hline \text { Distraction } & A & B & C & D & E & F & G & H & I \\ \hline \text { Driving Incidents } & 668 & 378 & 194 & 163 & 133 & 134 & 111 & 111 & 89 \\ \hline \end{array} $$ where \(A=\) Wireless device (cell phone, PDA) $$ \begin{aligned} B &=\text { Passenger } \\ C &=\text { Something inside } \mathrm{car} \\ D &=\text { Vehicle } \\ E &=\text { Personal hygiene } \\ F &=\text { Eating } \\ G &=\text { Something outside car } \\ H &=\text { Talking/singing } \\ I &=\text { Other } \end{aligned} $$ If an incident caused by a distraction is picked at random, what is the probability that it was caused by a. The use of a wireless device? b. Something other than personal hygiene or eating?

Step by step solution

01

Total number of incidents

To find the probability of an incident caused by a specific distraction, we first need to determine the total number of incidents. To do this, add all incidents together. Total number of incidents = 668 + 378 + 194 + 163 + 133 + 134 + 111 + 111 + 89

02

Calculate probabilities

Now that we have the total number of incidents, we can calculate the probability of each incident type. The probability can be calculated using the formula: \[P(\text{Distraction}) = \frac{\text{Number of Distractions of given type}}{\text{Total number of incidents}}\] Calculate required probabilities: a) The probability that an incident was caused by the use of a wireless device (A): \(P(A) = \frac{668}{\text{Total number of incidents}}\) b) The probability that an incident was caused by something other than personal hygiene (E) or eating (F): \(P(\text{Not E or F}) = 1 - P(\text{E or F})\) To find the probability of E or F, first find the sum of incidents caused by personal hygiene (E) and eating (F): Number of incidents caused by E or F = 133 + 134 Then, calculate the probability of E or F: \(P(\text{E or F}) = \frac{\text{Number of incidents caused by E or F}}{\text{Total number of incidents}}\) Now, calculate \(P(\text{Not E or F})\): \(P(\text{Not E or F}) = 1 - P(\text{E or F})\)

03

Calculate the values

Find the values for the total number of incidents, the probability of an incident caused by a wireless device, and the probability of an incident caused by something other than personal hygiene or eating: Total number of incidents = 668 + 378 + 194 + 163 + 133 + 134 + 111 + 111 + 89 = 1981 a) \(P(A) = \frac{668}{1981} \approx 0.337\) b) Number of incidents caused by E or F = 133 + 134 = 267 \(P(\text{E or F}) = \frac{267}{1981} \approx 0.135\) \(P(\text{Not E or F}) = 1 - 0.135 = 0.865\) The probabilities for the given scenarios are: a) The probability that an incident was caused by the use of a wireless device is approximately 0.337. b) The probability that an incident was caused by something other than personal hygiene or eating is approximately 0.865.

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Key Concepts

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

Driver Distraction Statistics

Understanding driver distraction statistics is crucial in highlighting the risks and consequences of divided attention while driving. Engaging in non-driving activities can significantly increase the risk of crashes and near-crashes.

The data from the study presented, encompasses various forms of driver distractions ranging from the use of wireless devices to eating and personal care. Incidents resulting from such distractions are meticulously listed, and it's evident that wireless devices top the list with 668 instances out of a total of 1981.

This statistical analysis serves as a foundation to grasp the real-world impact of each distraction type and underscores the importance of focused driving. When it comes to educational platforms, such data can also be used to craft exercises for students to apply mathematical concepts like probability, and understand their theoretical knowledge in practical, real-life situations.

Probability Calculation

Probability calculation is integral to predicting the likelihood of events based on quantitative data. In the context of the driver distraction statistics, probability calculation allows us to determine the chances of an incident being caused by a particular type of distraction.

To perform these calculations, a basic understanding of probability is required, which includes knowing how to sum up events and assess the total number of occurrences. With the exercise provided, we first sum all the incidents to get the total, a critical initial step as it sets the denominator for subsequent probability calculations.

Then, to find the probability of an individual distraction's role in incidents, one would divide the number of incidents caused by that specific distraction by the total number of incidents. This approach can be applied to various scenarios and is a fundamental skill in finite mathematics.

Probability Formula

The probability formula is a mathematical expression used to calculate the likelihood of an event occurring. It's elegantly simple and expressed as: \[P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\]In practice, the formula involves dividing the count of the event of interest by the total count of all possible outcomes. In the case of our driver distraction example, the number of incidents caused by wireless devices is divided by the total number of driving incidents to yield the probability of such an event.

Applying the formula to our textbook exercise, we understand not only how to find \(P(A)\), the probability of an incident caused by a wireless device, but also how to use the complement rule to find probabilities of combined events, such as the likelihood of an incident being due to factors other than personal hygiene or eating.