Question

Police often set up sobriety checkpointsroadblocks where drivers are asked a few brief questions to allow the officer to judge whether or not the person may have been drinking. If the officer does not suspect a problem, drivers are released to go on their way. Otherwise, drivers are detained for a Breathalyzer test that will determine whether or not they will be arrested. The police say that based on the brief initial stop, trained officers can make the right decision \(80 \%\) of the time. Suppose the police operate a sobriety checkpoint after 9 p.m. on a Saturday night, a time when national traffic safety experts suspect that about \(12 \%\) of drivers have been drinking. a) You are stopped at the checkpoint and, of course, have not been drinking. What's the probability that you are detained for further testing? b) What's the probability that any given driver will be detained? c) What's the probability that a driver who is detained has actually been drinking? d) What's the probability that a driver who was released had actually been drinking?

Short answer

a) \(0.20\), b) \(0.272\), c) \(0.353\), d) \(0.033\).

Step-by-step solution

Understanding the Problem

Let's define the events:- Let \( D \) be the event that a driver has been drinking. Thus, \( P(D) = 0.12 \) and \( P(D^c) = 0.88 \), the complement meaning the driver has not been drinking.- Let \( T \) be the event that the driver is detained for further testing.- The probability that a drinking driver is correctly detained is \( P(T | D) = 0.80 \).- The probability of incorrect detention for a non-drinking driver is \( P(T | D^c) = 1 - 0.80 = 0.20 \).

Calculating Probability of Being Detained if Sober

Given you have not been drinking \((D^c)\), the probability of being detained is:\[ P(T | D^c) = 0.20 \]

Calculating Probability of Any Driver Being Detained

We need to find \( P(T) \), the total probability of detention. Apply the law of total probability:\[ P(T) = P(T | D) P(D) + P(T | D^c) P(D^c) \]Substitute the known values:\[ P(T) = (0.80 \times 0.12) + (0.20 \times 0.88) = 0.096 + 0.176 = 0.272 \]

Calculating Probability of Drinking if Detained

We need to calculate \( P(D | T) \), the probability that a detained driver was actually drinking, using Bayes' theorem:\[ P(D | T) = \frac{P(T | D) \cdot P(D)}{P(T)} \]Substitute the known values:\[ P(D | T) = \frac{0.80 \times 0.12}{0.272} = \frac{0.096}{0.272} \approx 0.353 \]

Calculating Probability of Drinking if Released

We need to calculate \( P(D | T^c) \), the probability that a driver who is released was actually drinking.First find \( P(T^c) \), the probability of being released:\[ P(T^c) = 1 - P(T) = 1 - 0.272 = 0.728 \]Now find \( P(D | T^c) \) using complement rule and Bayes' theorem:\[ P(D | T^c) = \frac{P(T^c | D) \cdot P(D)}{P(T^c)} \]Where \( P(T^c | D) = 0.20 \), therefore:\[ P(D | T^c) = \frac{0.20 \times 0.12}{0.728} = \frac{0.024}{0.728} \approx 0.033 \]

Key concepts

Bayes' Theorem

Bayes' Theorem is a fundamental concept in probability theory. It provides a way to update our beliefs or probabilities about events based on new information. It's especially useful when we want to find the probability of a certain cause, given an observed effect.
In our exercise scenario, we're interested in knowing the probability that a detained driver is indeed drinking based on the information gathered at the checkpoint. Bayes' Theorem helps us here by relating conditional probabilities:
- We use the formula: \[ P(D | T) = \frac{P(T | D) \cdot P(D)}{P(T)} \]Here, - \(P(D | T)\) represents the probability that a driver is drinking given they've been detained.- \(P(T | D)\) is the probability of detaining someone who is drinking, which we know is 0.80 in this case.
- \(P(D)\) is the probability any driver is drinking (12%).- \(P(T)\) is the total probability of any driver being detained.
Applying Bayes' Theorem here shows us how strongly our evidence supports a conclusion—that a detained driver is drinking in this case.
Bayes' Theorem simplifies problems involving many conditions and dependencies by breaking them down into manageable parts and allowing sequential updating of probabilities as new data becomes available. It's a powerful and often intuitive tool in statistics.

Law of Total Probability

The Law of Total Probability is another cornerstone of probability theory. It helps in calculating the total probability of an event based on several different ways that event can occur. It's usually employed when there's a need to piece together individual probabilities from various scenarios.
In this exercise, we use this law to find the total probability that a driver is detained, denoted as \(P(T)\).
- According to the law, the probability of the detention event can be calculated by considering all possible underlying reasons for detention, which in this case are 'drinking' and 'not drinking': \[ P(T) = P(T | D) \cdot P(D) + P(T | D^c) \cdot P(D^c) \]This breaks down as:- \(P(T | D)\) is the probability they're detained given they've been drinking (80%).- \(P(D)\) reflects the chance that any driver has been drinking (12%).
- \(P(T | D^c)\) is the probability of detaining a nondrinker wrongly (20%).- \(P(D^c)\) is the probability a driver isn't drinking (88%).
Using the law, these factors combine to give us the overall probability of detention, helping us see how the layers of possibility stack up to create the total outcome. This full perspective is essential in calculating realistic expectations of outcomes based on complex scenarios.

Complement Rule

In probability, the Complement Rule is a simple yet powerful tool. It relates the probabilities of events and their complements, which are what do NOT happen versus what does.
This rule states that the probability of the complement of an event is 1 minus the probability of the event itself. Mathematically, it's expressed as: \[ P(A^c) = 1 - P(A) \]Where \(A^c\) is the complement of event A. This rule is particularly helpful when it’s simpler or more straightforward to calculate the probability of what doesn't occur.
In our checkpoint exercise, the complement rule is used to find the probability of release, which is the complement of being detained. Thus:- \(P(T^c) = 1 - P(T)\)- \(T^c\) is the event a driver is released.- From prior calculations, we have \(P(T) = 0.272\), making \(P(T^c) = 0.728\).
The complement rule helps assure we consider both sides of probability—sometimes understanding what 'doesn't' happen is as insightful as what 'does.' In this scenario, it's crucial to recognize the probability of wrongful release—drivers who might be drinking yet aren't detained. Such insights are essential to paint a full picture of the checkpoint's effectiveness.