Total Probability Theorem: Definition, Example, and Applications

Explore the total probability theorem in this comprehensive guide. Learn how to calculate the probability of an event based on different conditions, using the formula for the total probability theorem. Discover real-world applications in fields like medical diagnosis and finance, and see how the theorem relates to other concepts in probability theory, such as conditional probability and Bayes’ theorem. Whether you’re new to probability theory or looking to deepen your understanding, this article has everything you need to know about the total probability theorem.

Probability is the branch of Mathematics that describes chance and randomness. Simply, it estimates the likelihood of different outcomes based on the situation. It is the ratio of the number of favourable outcomes to the number of total outcomes.

In this article, we will briefly discuss one of the probability concepts, i.e., the Total Probability Theorem, which helps find the probability of an event based on its relationship to other events.

So, without further delay. Let’s explore the article.

Table of Content

• Total Probability Theorem
  • Definition
  • Proof
• Examples of Total Probability Theorem
• Properties
• Applications

What is Total Probability Theorem?

Let’s take a scenario where you have to predict the weather. So, what parameters will you use? It would be like Temperature, Humidity, Wind Speed, Wind Direction, Air Density, Precipitation, Pressure, etc.

Now, if you have all these parameters, how will you find the rainfall probability? Here, the total probability theorem comes into the picture. 

In short, Total Probability Theorem is a way to find the probability of something happening by looking at all the different ways it could happen.

Now, it’s time to define the Total Probability Theorem formally. It is also known as the Law of Total Probability.

Related Read – Probability and Non Probability Sampling

Statement of Total Probability Theorem

Let C1, C2, C3, C4…., form the partition of the sample space S; then we can calculate the probability of an event A as:

P(A) = ∑ P(A | Ci) * P(Ci), i = 0, 1, 2,…, n

or

P(A) =  ∑ P (A ∩ Ci)

Note: C1, C2, C3, …, and Cn should be Mutually Exclusive Events.

Proof of Theorem

Let S be a sample space and C1, C2, C3, …, and Cn be the set of mutually exclusive events of sample space S, i.e., 

E = C1 ሀ C2 ሀ C3 ሀ C4 ሀ …. ሀ Cn

C1 ∩ C2 ≠ 0

 Now, let’s consider event A such that

A = A ∩ S

=> A = A ∩ {C1 ሀ C2 ሀ C3 ሀ C4 ሀ …. ሀ Cn}

=> A = {A ∩ C1} ሀ  {A ∩ C2} ሀ  {A ∩ C3} ሀ {A ∩ C4} ሀ … ሀ{A ∩ Cn} 

Now, taking the probability on both the side, we get:

P(A) = P(A ∩ C1) + P(A ∩ C2) + P(A ∩ C3) + P (A ∩ C4) + … + P(A ∩ Cn) 

=> P(A) = P(C1) * P (A | C1) +  P(C2) * P (A | C2) + P(C3) * P (A | C3) + …. + P(Cn) * P (A | Cn) 

The above calculation is done using the formula of Conditional Probability.

=> P(A) =∑ P (Ci) * P (A | Ci)      i = 1, 2, 3, …, n

Examples of Total Probability Theorem

Ques – 1: A company has three factories (F1, F2, and F3) that produce a certain product, and the probability that each factory produces a defective item is 0.2, 0.3, and 0.4, respectively.

What is the probability that a randomly selected item is defective?

Solution: Let E be the event that the selected item is defective.

Now, using the total probability theorem, we get

P(E) = P(E|F1) * P(F1) + P(E|F2) * P(F2) + P(E|F3) * P(F3)

=> P(E) = (0.2) (1/3) + (0.3) (1/3) + (0.4) * (1/3)

=> P(E) = 0.3

Hence, the probability that any randomly selected item is defective is 0.3.

Ques – 2: A coin is tossed three times.

If the outcome on the first toss is Head, then the probability of getting the head on the second toss is 0.6. If the outcome on the first toss is Tails, then the probability of getting the heads on the second toss is 0.4.

What is the probability of getting exactly two heads in the three tosses?

Solution: Let E1, E2, and E3 be three events in the first, second, and three tosses that result in heads, respectively.

P (E1 ∩ E2 ∩ H3′) = P(H1) * P(H2 | H1) * P(H3′ | H1 ∩ H2) + P( H1′) P(H2 | H1′) * P (H3 | (H1 ∩ H2)’)

=> P (E1 ∩ E2 ∩ H3′) = (1/2)*(0.6)*(1/2) + (1/2)*(0.4)*(1/2)

=> P (E1 ∩ E2 ∩ H3′) = 0.25

Properties of Theorem of Total Probability

• Theorem applies only to Mutually Exclusive (i.e., no common element in two sets) and Exhaustive Events (i.e., the union of all subsets is a sample space).
• It is used to calculate the probability of an event given any number of related events.
• Used with Baye’s Theorem to update the probability of a hypothesis given new evidence.
• Used to calculate 
  • the expected value, given the probabilities of related events and 
  • conditional probabilities, which are probabilities that depend on other events.

Application of Theorem of Total Probability

The Theorem of Total Probability can be used in Statistics, Engineering, Economics, and Physics.

Let’s explore how this theorem is used.

Disease Diagnosis: It helps to calculate the probability of a patient having a disease, given the prevalence of the disease and the accuracy of the disease.

Investment Analysis: Used to calculate the expected return on an investment, given the probability of different economic outcomes.

Marketing: Used to calculate the probability of a particular event occurring, given the probability of different scenarios leading up to that event.
Game Theory: Used to analyze the strategies of different players in a game, given the probability of different outcomes.

Conclusion

In this article, we have briefly discussed total probability theorem, how to use it with the help of example. Later in the article, we have also covered its properties and application.

Hope you will like the article,

Happy Learning!!