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Bayes' Theorem

Chapter 5: Bayes’ Theorem Breathalyzer Example

You are a police officer in Baltimore and it’s New Year’s Eve. As usual, roadblocks are set-up at various points throughout the city to combat drunk driving. Throughout the evening you and your fellow officer are giving random drivers breathalyzer tests. To your surprise, the night is going well and you’ve had few incidents. Around 2 am you randomly pull over a vehicle and have the driver take a breathalyzer test, and the result is positive. You assume the test is accurate and think nothing of it as you process the driver.

After your shift ends early that morning you are talking with your partner. She doesn’t believe that the breathalyzer tests are anywhere near accurate. You ask her why and she tells you some stats: it is true that the breathalyzer always detects a truly drunk person, but only 1 in 1000 drivers are typically drunk. What’s more, the probability of the test being positive is only 8%. You shake your head while you try to put together what she has said. You’ve never questioned the test before but now you are.

What should you believe? Let’s pull this scenario apart.

To clarify once more, Bayes’ Theorem helps us update a hypothesis based on new evidence. We won’t restate this for the remaining scenarios but thought it would be helpful to point it out one last time.

In this breathalyzer scenario, your hypothesis is that the person is drunk and your evidence is a positive breathalyzer test. Originally, you never questioned the accuracy of the test. You thought it was accurate since it has a 100% success rate of always detecting a truly drunk person. But after listening to your partner you are now questioning this belief.

As in our flu example above, this is where Bayes’ Theorem comes in and helps us have a better understanding of probability. In this scenario, we are given two additional pieces of information that can help us come to a more precise probability. Let’s review those quickly before we move on.

  • We know that 100% of the time the breathalyzer test will give a positive result for a truly drunk driver. By truly drunk we mean that a blood test would confirm that the person is over the blood alcohol limit.
  • We know that 1 in 1000 drivers drive drunk, so the probability of any driver being drunk is 0.1%* (we calculated this by dividing 1/1000 and then multiplying by 100).
  • We also know that the breathalyzer test will give a positive result 8% of the time regardless if it is accurate or not.

To start, we always need to determine what we are wanting to find. We want to know the probability of someone actually being drunk given that the breathalyzer test is positive.

Perfect. We now know what we are solving for, so let’s move on and tackle it in a few ways.
We are going to look at this problem through two different lenses. A) Visualize the Problem: we will visualize the problem by using a Venn diagram. B) Plugging Into Bayes’ formula and Solving: we will solve the problem by plugging our numbers into the Bayes’ Theorem formula.

Example 2.1 Visualize the Problem

Let’s visualize with a Venn diagram.

Circle #1: The area inside this circle represents all possible outcomes. In this scenario, it represents all people who could be drunk while driving. The small circle labeled “A” represents the .1% of drivers who actually are drunk. “A” is an event, and its probability is .1%. This probability is represented in our formula as P(A).Venn Diagram Bayes Theorem Drunk Drivers Example 1

Circle #2: The area inside this circle represents all possible outcomes. In this scenario, it represents all possibilities for the breathalyzer test. The small circle labeled “B” represents the 8% of the tests that are positive.“B” is an event, and its probability is 8%. This probability is represented in our formula as P(B).Venn Diagram Bayes Theorem Drunk Drivers Example 2

Circle #3: This is where all the pieces come together. In this circle, we have combined both events “A” and “B”.

Here is how the entire visual can be understood:

  • The white area inside this circle represents people who are not drunk drivers and breathalyzer tests that are negative.
  • The area where only Circle A covers shows us people who are drunk while driving.
  • The area where only Circle B covers shows us the total amount of breathalyzer tests that are positive.
  • Boom! Take a look at the dark area where the two circles overlap. This is what we are really interested in! This is our question that we want to be answered but in visual form. We want to know the probability P(A|B) of a driver being drunk given that the breathalyzer test is positive. This probability is found where both events occur together and is called an intersection.

Venn Diagram Bayes Theorem Drunk Drivers Example 3

With both circles now merged, we can visually see our question and what we are trying to solve for. Although we won’t be solving the question with a Venn diagram, the diagram does help us visualize what we are trying to understand. If P(A) is the probability a driver driving drunk, and P(B) is the probability a breathalyzer test being positive, what is the probability of both?

Example 2.2 Plugging Into Bayes’ Formula and Solving

Let’s follow our four steps again. To make things clear, we’ll clarify what we want to find.

Step 1: Determine what you want to find. We want to know the probability of someone being truly drunk given that the breathalyzer test is positive.

Step 2: Write the above as a formula. Let’s translate what we are solving for into the formula. In other words, we’ll bring the language of Step #1 above into the formula.

Here is Bayes’ formula:
Bayes Theorem Formula

Now, let’s translate with what we are solving for.Bayes Theorem Formula for Example Drunk

Step 3: Find each ingredient and label it. From the scenario we know the following:

  • P(A) – In our formula, this ingredient is represented as P(Drunk) and answers the question: What is the probability of a driver being drunk? This number is .001
  • P(B|A) – In our formula, this ingredient is represented as P(Positive|Drunk). This number is 1, not .1, but 1 as in 100%.
  • P(B) – In our formula, this ingredient is represented as P(Positive) and answers the question: What is the probability of a breathalyzer test being positive? This number is .08

Step 4: Plug each ingredient into the formula and solve.
Bayes Theorem Example Math Solved Drunk

Conclusion: So, after all the work and plugging each ingredient into the formula our answer is 1.25%. We can conclude from this that the probability of a driver having a positive test and actually being drunk is 1.25%. In other words, for every 1000 drivers being tested we have the following:

  • We have 1 truly drunk driver who tested positive with the breathalyzer.
  • We have roughly 79 other drivers who also tested positive with the breathalyzer but are not drunk.
  • In total we have 80 drivers testing positive, and the probability of one of them testing positive and truly being drunk is around 1.25%.

That’s an eye-opener! Remember what Bayes’ Theorem does: it can help us quantify skepticism and enable us to have a clearer understanding. Originally, we thought the probability of the driver being drunk was quite high, but now we see it is only around 1.25%.

In this scenario, the police officer would now take a very different view of breathalyzer tests and would likely be much more skeptical of their accuracy. *As a side note, there are other complexities to situations like this, such as the test actually being random, etc. Did the police officer pull the vehicle over because of how the driver was driving? Did the officer provide a breathalyzer test because if how they were acting? Responding? New information such as this would impact the entire equation. For sake of ease and teaching we have kept the problem very basic.

Go to Chapter 6: Bayes’ Theorem Peacekeeping Example.