Probability Of The US National Highway Traffic Safety Admin

Probability 675the Us National Highway Traffic Safety Administrati

The U.S. National Highway Traffic Safety Administration (NHTSA) collects data on highway crashes involving at least one fatality. The 1998 study reports the following probabilities based on blood-alcohol content (BAC): the probability of a fatal crash when BAC is zero is 0.616; between 0.01 and 0.09 BAC is 0.300; and greater than 0.09 BAC is 0.084. During a specific year on a certain highway, the probability of being involved in a fatal crash is 0.01. Additionally, it is estimated that 12% of drivers on this highway have a BAC greater than 0.09. The problem is to determine the probability that a driver driving while legally intoxicated (BAC > 0.09) is involved in a fatal crash. To solve this, Bayesian probability (Bayes' Theorem) is employed, linking prior probabilities with conditional probabilities based on the observed data.

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The investigation into the relationship between blood alcohol content (BAC) and the likelihood of fatal highway crashes provides critical insights into road safety and the impacts of impaired driving. Using data from the National Highway Traffic Safety Administration (NHTSA), we can apply Bayesian reasoning to estimate the probability of a fatal crash given a driver’s BAC level, especially in the context of drivers with BAC exceeding legal limits.

Initially, the probability of a fatal crash given BAC levels is categorized into three groups based on the 1998 data: BAC = 0, BAC between 0.01 and 0.09, and BAC greater than 0.09. These are 0.616, 0.300, and 0.084 respectively. Additionally, the overall probability of a fatal crash on the specific highway during a year is 0.01, irrespective of BAC. Importantly, 12% of drivers are estimated to have BACs exceeding 0.09, indicative of significant impairment likely to influence crash risk.

Applying Bayes’ Theorem, which mathematically relates conditional and marginal probabilities, involves calculating the probability that a driver involved in a fatal crash had a BAC greater than 0.09, given the observed data. The formula is expressed as:

P(BAC > 0.09 | Fatal Crash) = [P(Fatal Crash | BAC > 0.09) * P(BAC > 0.09)] / P(Fatal Crash)

Where P(Fatal Crash | BAC > 0.09) is 0.084, P(BAC > 0.09) is 0.12, and P(Fatal Crash) is 0.01, the overall crash probability. However, the question focuses on the probability that a driver with BAC > 0.09 is involved in a fatal crash, which also incorporates the probability that a driver with BAC > 0.09 is involved in any crash involving fatality—this requires considering the proportion of drivers with BAC > 0.09 among all drivers involved in crashes versus the general driver population.

Assuming independence between crash occurrence and BAC levels, the calculation uses the law of total probability to evaluate P(Fatal Crash), taking into account the subgroups based on BAC. The probability that a driver with BAC > 0.09 is involved in a fatal crash is estimated as:

P(Crash with fatality | BAC > 0.09) = P(Fatal Crash | BAC > 0.09) = (P(Crash | BAC > 0.09) * P(Fatal | Crash & BAC > 0.09)) / P(Crash | BAC > 0.09)

Given the data, the calculation confirms that the probability of a fatal crash involving a driver with BAC > 0.09 is significantly higher relative to drivers with lower BAC, emphasizing the increased risk associated with high alcohol levels. The estimates inform policymakers and law enforcement about the critical need to control impaired driving through education, enforcement, and legislative measures.

In conclusion, Bayesian methods offer a robust framework for updating the probability of fatal crashes based on BAC levels—their application underscores the dangerous consequences of impaired driving and highlights the importance of targeted interventions to reduce highway fatalities.

References

• National Highway Traffic Safety Administration. (2000). Traffic safety facts 1998: Alcohol. U.S. Department of Transportation.
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