Theorem of Total Probability

Suppose we are in a situation where all events in a sample space are mutually exclusive and exhaustive.

Mutually exclusive means the outcomes are separate from each other, like each time you roll a die.

Exhaustive means all outcomes are accounted for. (You will see in the example)


Example.

Let us say a company buys parts from 3 other companies.

It gets
60% from company A
40% from company B
20% from company C


Company A ships defective parts 1% of the time (0.01)
Company B ships defective parts 5% of the time (0.05)
Company C ships defective parts 10% of the time (0.10)

What are the odds of the company buying a defective part?

Now we have a situation that is exhaustive since all 3 companies comprise 100% of the outcomes. It is also mutually exclusive since one company doesn't affect the other and is "separate".

The theorem of total probability states that when we have an event that is mutually exclusive and exhaustive it can be found by adding the combination of disjoint outcomes. That is to say looking at each company separately and adding them together.
The way to write this is

P(defective part is bought) = P(defective shipped from A) + P(defective shipped from B) + P(defective shipped from C). This represents a "total" account of all outcomes.

The math formula looks like
P(F) = P(F and E₁) + (F and E₂) + (F and E₃) + ...(as many as it takes to get "total" probability. "and" in the formula can me taken to mean multiplication or times.)

The P(defective from a company) is the portion bought from any company times(x) the chance the chance sends something defective.

P(A defective) = 0.6*0.01 = 0.006
P(B defective) = 0.4*0.05 = 0.02
P(C defective) = 0.2*0.10 = 0.02

So
P(defective part is bought) = 0.006 + 0.02 + 0.02 = 0.046 = 4.6%