Law of total probability

# Law of total probability | Engineering Mathematics - Civil Engineering (CE) PDF Download

Given n mutually exclusive events A1, A2, …Ak such that their probabilities sum is unity and their union is the event space E, then Ai ∩ Aj = NULL, for all i not equal to j, and
A1 U A2 U ... U Ak = E
Then Total Probability Theorem or Law of Total Probability is:

where B is an arbitrary event, and P(B/Ai) is the conditional probability of B assuming A already occured.

Proof:
Let A1, A2, …, Ak be disjoint events that form a partition of the sample space and assume that P(Ai) > 0, for i = 1, 2, 3….k.
such that:
A1 U A2 U A3 U ....U AK = E(Total)
Then, for any event B, we have,
B = B ∩ E
B = B ∩ (A1 U A2 U A3 U ....U AK)
As intersection and Union are Distributive. Therefore,
B = (B ∩ A1) U (B ∩ A2)U ... U(B ∩ AK)
Since all these partitions are disjoint. So, we have,
P(B ∩ A1) = P(B ∩ A1) U P(B ∩ A2)U ... U P(B ∩ AK)
That is, addition theorem of probabilities for union of disjoint events.
Using Conditional Probability
P(B / A) =  P(B ∩ A) / P(A)
Or by the multiplication rule that,
P(B ∩ A) = P(B / A) x P(A)
Here events A and B are said to be independent events if P(B|A) = P(B), where P(A) not equal to Zero(0),
P(A ∩ B) = P(A) * P(B)
where P(B|A) is the conditional probability which gives the probability of occurrence of event B when event A has already occurred. Hence,
P(B ∩ Ai) = P(B | Ai).P(Ai) ; i = 1, 2, 3....k
Applying this rule above we get,

This is the law of total probability.
The law of total probability is also referred to as total probability theorem or law of alternatives.

Note: The law of total probability is used when you don’t know the probability of an event, but you know its occurrence under several disjoint scenarios and the probability of each scenario.

Application: It is used for evaluation of denominator in Bayes’ theorem.

Example: We draw two cards from a deck of shuffled cards with replacement. Find the probability of getting the second card a king.
Solution: Let,
A – represent the event of getting the first card a king.
B – represent the event that the first card is not a king.
E – represent the event that the second card is a king.
Then the probability that the second card will be a king or not will be represented by the law of total probability as:

P(E) = P(A)P(E | A) + P(B)P(E | B)
Where,
P(E) is the probability that second card is a king,
P(A) is the probability that the first card is a king,
P(E | A) is the probability that the second card is a king given that first card is a king,
P(B) is the probability that the first card is not a king,
P(E | B) is the probability that the second card is a king but the first card drawn is not a king.
According to question:
P(A) = 4 / 52
P(E | A) = 4 / 52
P(B) = 48 / 52
P(E | B) =  4 / 52
Therefore,
P(E)
= P(A)P(E | A) + P(B)P(E | B)
=(4 / 52) * (4 / 52) + (48 / 52) * (4  / 52)
= 0.0769230

The document Law of total probability | Engineering Mathematics - Civil Engineering (CE) is a part of the Civil Engineering (CE) Course Engineering Mathematics.
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## FAQs on Law of total probability - Engineering Mathematics - Civil Engineering (CE)

 1. What is the law of total probability?
Ans. The law of total probability is a fundamental concept in probability theory. It states that for any event A and a set of mutually exclusive events B1, B2, ..., Bn that partition the sample space, the probability of event A can be calculated by summing the conditional probabilities of A given each of the events B1, B2, ..., Bn, weighted by their respective probabilities.
 2. How is the law of total probability applied in real-world scenarios?
Ans. The law of total probability is commonly used to solve problems involving conditional probabilities in real-world scenarios. For example, it can be used to determine the probability of a particular medical condition given different test results, or the probability of a customer purchasing a product based on different demographic factors.
 3. Can you provide an example to illustrate the law of total probability?
Ans. Certainly! Let's say we have two factories, Factory A and Factory B, that produce a certain product. Factory A produces 60% of the total output, while Factory B produces the remaining 40%. The defect rates for Factory A and Factory B are 5% and 10% respectively. If a randomly selected product is found to be defective, what is the probability that it came from Factory A? To solve this, we can apply the law of total probability. We calculate the probability that the product came from Factory A given that it is defective by multiplying the probability of Factory A producing a defective product (0.05) by the probability of selecting a product from Factory A (0.6). We then divide this by the sum of the probabilities of both factories producing a defective product (0.05 * 0.6 + 0.1 * 0.4). The result is the probability that the product came from Factory A given that it is defective.
 4. Are there any limitations or assumptions associated with the law of total probability?
Ans. Yes, there are a few limitations and assumptions associated with the law of total probability. One assumption is that the events B1, B2, ..., Bn must be mutually exclusive and exhaustive, meaning that they cannot overlap and they must cover the entire sample space. Additionally, the law of total probability assumes that the conditional probabilities of event A given each of the events B1, B2, ..., Bn are well-defined and can be accurately determined.
 5. How does the law of total probability relate to Bayes' theorem?
Ans. The law of total probability is closely related to Bayes' theorem, which is another important concept in probability theory. Bayes' theorem allows us to update our prior beliefs about the probability of an event based on new evidence. The law of total probability is often used in the derivation of Bayes' theorem. By applying the law of total probability, we can calculate the probabilities needed to apply Bayes' theorem and make more informed probabilistic judgments based on new information.

## Engineering Mathematics

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