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Bayes' Theorem

Chapter 6: Bayes’ Theorem Peacekeeping Example

You are a soldier and have recently shipped out across the Atlantic on your fourth peacekeeping tour. A few weeks into your mission you are on patrol and see an injured family across the road from you. You are about to go to them when suddenly there is a surprise attack and you find yourself pigeonholed against a burnt-out vehicle. You stop to listen and are suddenly filled with horror as you see a truck turning the corner. There is no doubt it is an enemy vehicle, but you didn’t have time to see if the truck was rigged with a gunner on the back and if there is you don’t want to be caught in the open.

You quickly do some mental calculations and recall what you learned in your debrief. The rebels have roughly 54 dilapidated trucks and 22 of them are rigged with guns in the back. Rebels in a truck are one thing, but rebels in a truck rigged with a gun on the back? You don’t want to be caught in the open with that.

You pop your head out to get a better look and a wave of bullets hits the vehicle in front of you. The rebel truck is now about 150 yards away, but you are still uncertain if the shots came from the truck or somewhere else. If the truck is rigged with a gun, the chance of it having fired at you is pretty high, maybe at 80%.

You continue to think. Considering how heavy the firepower was and the environment you are in, you peg the possibility of being shot at 50%. What should you do? Should you risk crossing the street to help the family?

Let’s break this scenario apart.

As in our other scenarios, let’s break this scenario apart to see exactly what we are dealing with.

In this scenario, your hypothesis is that the truck is rigged with a gun and your evidence is the wave of bullets. Your initial assumption is that the truck is likely rigged with a gun since there is an 80% chance it would have fired at you if it was rigged. But is that right? Are you making an accurate conclusion?

Again, this is where Bayes’ Theorem can help us better understand the situation and make a more informed decision. In the scenario, we are given two additional pieces of information that can help us come to a more precise probability of the truck being rigged with a gun given the intense wave of bullets. Let’s take a moment to review:

  • We know that the probability of the truck firing at you if it is rigged with a gun is 80%.
  • We know the probability of the truck being rigged with a gun is 40%* (we calculated this by dividing 22 rigged trucks by 54 total trucks. 22/54 = ~40%.)
  • We know that the probability of a rebel having heavy firepower is 50%.

To start, we always need to determine what we are wanting to find.

We want to know the probability of the truck being rigged with a gun given that we were just fired at with heavy firepower.

Visualize the Problem – Let’s visualize with a Venn diagram

Circle #1: The area inside this circle represents all possible outcomes. In this example, the area represents all rebel trucks. The shaded circle labeled “A” represents the 40% of all rebel trucks that are rigged with a gun. * To get this number we simply divide the number of rigged trucks (22) by the number of total trucks (54). The answer is ~40%. “A” is an event, and its probability is 22%. This probability is represented in our formula as P(A).

Circle #2: The area inside this circle also represents all possible outcomes. In this instance it represents firepower. The shaded circle labeled “B” represents the 50% of rebels who could shoot you with very heavy firepower. What this means is that within the entire circle there are two possible outcomes: rebels either have heavy firepower or do not have heavy firepower. “B” is an event, and its probability is 50%. This probability is represented in our formula as P(B).

Circle #3: Alright. Let’s take a look at how these two events combined. This is where the magic happens!
Here is a quick breakdown of how you can understand the entire diagram:

  • The white area inside this circle represents rebel trucks that are not rigged with a gun and rebels who do not have heavy firepower.
  • The area covered by Circle A shows us the total amount of rebel trucks that are rigged with a gun.
  • The area covered by Circle B shows us rebels who have heavy firepower capability.
  • Now, move your eyes to the dark area where the two circles overlap. This is what we are really interested in! This is our original question in visual form. We want to know the probability P(A|B) of a rebel truck being rigged with a gun given heavy firepower. In other words, if we are in Circle B, what is the probability of being in A as well? This probability is found where both events occur together and is called an intersection.

With both circles now merged, we can visually see our question and what we are trying to solve for. If P(A) is the probability of a rebel truck being rigged with a gun, and P(B) is the probability of heavy firepower, what is the probability of you being where they overlap – and both events occurring at the same time?

Plugging Into Bayes’ formula and Solving

To solve by using Bayes’ Theorem we’ll follow four steps. For sake of ease, we’ll begin by re-stating what we are wanting to find.

Step 1: Determine what you want to find. We want to know the probability of the truck being rigged with a gun given that we were just fired at with heavy firepower.

Step 2: Write the above as a formula. Here is Bayes’ formula:

Now, let’s translate with what we are solving for.

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(Rigged) and answers the question: What is the probability of a rebel truck being rigged with a gun? This number is .4
  • P(B|A) – In our formula, this ingredient is represented as P(Firepower|Rigged). This number is .8
  • P(B) – In our formula, this ingredient is represented as P(Firepower) and answers the question: What is the probability of a rebel having heavy firepower? This number is .5

Step 4: Plug each ingredient into the formula and solve.

Conclusion: Once we’ve plugged all of our ingredients into the formula we arrive at 64%. Based on this, we can conclude that the probability of the truck being rigged with a gun given that heavy firepower came our way is 64%. We originally thought the probability was higher around the 80% mark, but now we can see it hovers about 15 % lower.

If your original instinct was to stay behind the burnt out vehicle, knowing this probably won’t change that. A difference of 15% is not that great. But, if the result (posterior probability) had been lower around 20% or 30% instead of 64%, that might have changed your actions.

We chose this last scenario to contrast the first two and demonstrate that using Bayes Theorem does not always provide a clear answer. Sometimes a probability only slightly changes.

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