Impara Law of Total Probability | Probability of Complex Events

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

Let's provide some explanations:

We have split our space of elementary events into n different incompatible events;
We want to calculate the probability of some other event in this space of elementary events;
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:


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# Probability of selecting Product 1 and Product 2
P_H1 = 0.6
P_H2 = 0.4

# Defect rates for Product 1 and Product 2
P_A_cond_H1 = 0.1
P_A_cond_H2 = 0.05

# Calculate the overall probability of selecting a defective product
P_A = P_A_cond_H1 * P_H1 + P_A_cond_H2 * P_H2

# Print the results
print(f'The overall probability of selecting a defective product is {P_A:.4f}')

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Sezione 2. Capitolo 4

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