Problem 11 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

A breath analyzer, used by the police to test whether drivers exceed the legal limit set for the blood alcohol percentage while driving, is known to satisfy $$ \mathrm{P}(A \mid B)=\mathrm{P}\left(A^{c} \mid B^{c}\right)=p, $$ where \(A\) is the event "breath analyzer indicates that legal limit is exceeded" and \(B\) "driver's blood alcohol percentage exceeds legal limit." On Saturday night about \(5 \%\) of the drivers are known to exceed the limit. a. Describe in words the meaning of \(\mathrm{P}\left(B^{c} \mid A\right)\). b. Determine \(\mathrm{P}\left(B^{c} \mid A\right)\) if \(p=0.95\). c. How big should \(p\) be so that \(\mathrm{P}(B \mid A)=0.9\) ?

Step-by-Step Solution

Answer

(a) Probability of a false positive; (b) 0.95; (c) p = 0.9947.

Step 1: Understand the problem

We have events for a breath analyzer test: - Event \(A\): Breath analyzer indicates the legal limit is exceeded. - Event \(B\): Driver's blood alcohol percentage actually exceeds the legal limit. - \(\mathrm{P}(A \mid B)=\mathrm{P}(A^c \mid B^c)=p\) means the probability that the analyzer is correct, given the driver's true status. - \(\mathrm{P}(B)=0.05\) implies 5% of drivers exceed the limit.

Step 2: Explain the meaning of \(\mathrm{P}(B^c \mid A)\)

\(\mathrm{P}(B^c \mid A)\) is the probability that a driver's blood alcohol percentage does not exceed the legal limit, given that the breath analyzer indicates the legal limit is exceeded. Essentially, it measures the likelihood of a "false positive" result.

Step 3: Use Bayes' Theorem to solve part (b)

We need to find \(\mathrm{P}(B^c \mid A)\). Using Bayes' Theorem,\[\mathrm{P}(B^c \mid A) = \frac{\mathrm{P}(A \mid B^c) \cdot \mathrm{P}(B^c)}{\mathrm{P}(A)}\]We know \(\mathrm{P}(A \mid B^c)=1-p=0.05\) and \(\mathrm{P}(B^c)=0.95\). We also need \(\mathrm{P}(A)\). \[\mathrm{P}(A) = \mathrm{P}(A \cap B) + \mathrm{P}(A \cap B^c)\] \[\mathrm{P}(A) = \mathrm{P}(A \mid B) \cdot \mathrm{P}(B) + \mathrm{P}(A \mid B^c) \cdot \mathrm{P}(B^c)\] \[\mathrm{P}(A) = 0.95 \times 0.05 + 0.05 \times 0.95 = 0.05\] Therefore,\[\mathrm{P}(B^c \mid A) = \frac{0.05 \times 0.95}{0.05} = 0.95\]

Step 4: Solving part (c) using conditional probabilities

We seek \(p\) such that \(\mathrm{P}(B \mid A) = 0.9\). Using Bayes' Theorem,\[\mathrm{P}(B \mid A) = \frac{\mathrm{P}(A \mid B) \cdot \mathrm{P}(B)}{\mathrm{P}(A)}\] We need \(\mathrm{P}(A)\):\[\mathrm{P}(A) = 0.9 \times 0.05 + (1-p) \times 0.95\] Solving \(\frac{0.9 \times 0.05}{\mathrm{P}(A)} = 0.9\) for \(p\):\[0.9 \times \mathrm{P}(A) = 0.045\] Substitute \(\mathrm{P}(A)=0.045+(1-p)\times0.95\) and solve for \(p\):\[0.9(0.045+(1-p)\times0.95) = 0.045 \]After simplification:\[0.0405 + 0.855(1-p) = 0.045\]Solve:\[0.855\times(1-p) = 0.0045 \] \[1-p = \frac{0.0045}{0.855} \] \[ p = 1 - \frac{0.045}{0.855} \] Solve for \(p\):\[p = 1 - 0.005263 \] \[ p = 0.9947 \]

Step 5: Conclusion

By increasing the accuracy of the breath analyzer to \(p=0.9947\), \(\mathrm{P}(B \mid A)\) can be elevated to 0.9.

Key Concepts

Conditional Probability

Conditional probability is a concept used to determine the probability of an event occurring, given that another event has already taken place. In our scenario with the breath analyzer test, it helps us figure out the chance the test is accurate or not. Specifically,

• Event A represents a situation where the breath analyzer indicates that the legal limit is exceeded.
• Event B describes the actual state where the driver's blood alcohol percentage indeed exceeds the legal limit.

The notation \( \mathrm{P}(A \mid B) \) stands for the probability that the analyzer indicates the limit exceeded, given that the driver truly has exceeded the limit. In essence, it tells us how likely it is the analyzer gets it right when the driver is indeed over the limit. Bayes' Theorem then becomes an essential tool, as it offers a method to compute these probabilities based on known values. This theorem connects conditional probabilities, transforming our understanding of certain probabilities by considering the presence of others.

False Positive

In the context of testing and probability, a "false positive" occurs when a test incorrectly indicates the presence of a condition or characteristic. In the breath analyzer exercise,

• A false positive happens when the test shows that a driver's blood alcohol exceeds the legal limit when, in reality, it does not.
• This is expressed mathematically as \( \mathrm{P}(B^c \mid A) \), indicating the probability of the driver not exceeding the limit, given the analyzer shows they did.

Understanding false positives is critical because they can lead to unnecessary legal actions or stress. In practical terms, it's essential for tests like breath analyzers to minimize these occurrences to increase reliability and maintain trust in their results. Unfortunately, no test is perfect; thus, exploring how often these inaccuracies occur allows us to understand and improve test designs and accuracies.

Probability Distributions

Probability distributions play a crucial role in understanding the likelihood of various outcomes in random processes or experiments. In the breath analyzer scenario,

• The probability distribution gives us a detailed picture of how likely different results are, showing how probabilities are spread over different possible outcomes.
• For instance, knowing that 5% of drivers exceed the legal limit (\( \mathrm{P}(B) = 0.05 \)), helps us calculate and comprehend other probabilities linked to this scenario.

By using equations like \( \mathrm{P}(A \mid B) = 0.95 \) and \( \mathrm{P}(A \mid B^c) = 0.05 \), we set up the foundation for the probability of different states, and thus, their distribution. This information not only helps solve specific problems like calculating other conditional probabilities through Bayes' Theorem but also assists in designing tests with optimal accuracy in general practical applications.