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:
- 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:
12345678910111213# 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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Law of Total Probability
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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:
12345678910111213# 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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