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Bayes' Theorem

Chapter 7: No P(B) Provided and What Are You Looking For?

Before we jump into chapter 8 and are forced to discover P(B), it might be helpful to know how to read and understand scenarios. If you feel confident with this, please skip this chapter and move on. However, basic skills in this area are often overlooked and not taught properly. So, if you are keen to brush up on how to understand scenarios, keep reading!

In past chapters we dealt with scenarios where all of the ingredients were provided and labeled for us. This made plugging the numbers into Bayes’ formula quite effortless. Plus, we also defined what we were looking for.

The truth is though, scenarios don’t always give you everything so easily.

Sometimes merely understanding what you are looking for and defining ingredients is difficult enough. So, on this page, we’ll take a quick detour and provide you with a few tips on how to do these things.

Discovering What You Are Looking For: P(A|B)

This is the first and most important step you’ll take when solving a question. Before you can solve a question, you need to know what you are solving for. Here are three steps that can help you do this:

Step 1: Ignore all the extra filler provided in the question and always begin by writing down each statistic given. Doing this will help you focus on what matters most and disregard the rest. Be sure to give each statistic a label. For example:

  • 100% – positive breathalyzer tests for truly drunk drivers
  • .1% – the number of drivers who drive drunk
  • 8% – positive results given by the breathalyzer

Step 2: Look at what you’ve collected in the list above. For the majority of scenarios, you will have two or three numbers that are provided. Use these to help you brainstorm questions. Some great questions to ask are:

  • What probability am I wanting to know?
  • What belief is being questioned?

Again, ignore all the extra filler in the question (by filler we mean the backstory and other details that don’t have anything to do with the numbers above).

Step 3: Working with Step 2, begin to write down what you think you are trying to solve for. Remember, you can write your question out rough and then refine it (as in our example below). And always remember to include the terms probability and given that. Doing this will help you define each ingredient with confidence.

  • Version 1: If a breathalyzer is positive, what are the chances someone is drunk?
  • Version 2: What is the probability of someone being drunk if the breathalyzer is positive?
  • Version 3: What is the probability of a driver being drunk given that their breathalyzer test is positive?

Defining P(B), P(A) P(B|A)
Once we know what we are looking for, defining the various ingredients is straightforward. It is also much easier if you have used the terms probability and given that in Step 3 above.

Here’s how to do it:
Step 1: Write out what you are searching for. Example: What is the probability of a driver being drunk given that their breathalyzer test is positive?

Step 2: Label each ingredient:

  • Event “A” is always found between probability and given. In this case, it is Driver Being Drunk. The probability of this is P(A).
  • Event “B” is always found after given that. In this case, it is Breathalyzer Test is Positive. The probability of this is P(B).
  • P(B|A) is a combination of the above. In this case, P(B|A) is: The probability of a breathalyzer test is positive, given that a driver is drunk.
  • P(A|B) is also a combination of the above. In this case, P(A|B) is: The probability of a driver being drunk given that their breathalyzer test is positive.

Once you understand what you are looking for and have defined your ingredients, you are on your way to solving the question. That’s all there is to it!

Go to Chapter 8: No P(B) Provided – Bayes’ Theorem Flu Example.