Law of Total Probability

The law of total probability is a fundamental concept in probability theory. This law can be formulated as follows:

Let's provide some explanations:

1. We have split our space of elementary events into n different incompatible events;
2. We want to calculate the probability of some other event in this space of elementary events;
3. We can calculate P(A) using the formula described above.

This law is often used when a stochastic experiment can be divided into different stages, and each stage is stochastic too.

Example

Let's consider an example involving a manufacturing company that produces two types of products: Product 1 and Product 2.
The company produces 60% of Product 1 and 40% of Product 2.
The defect rate for Product 1 is 10%, while the defect rate for Product 2 is 5%. We want to calculate the probability of randomly selecting a defective product from the company's inventory.

In this example:

Event A: Selecting a defective product.
Partition events: H₁ = Selecting Product 1, H₂ = Selecting Product 2.
Now we can use the law of total probability to solve this task: