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# Mathematics | Law of total probability

The Law of total probability is important to find the probability of an event happening. If the probability of an event is going to happen is known to be 1 also for an impossible event it is likely to be 0. A fundamental rule in the theory of probability that is interconnected to marginal probability and conditional probability is called the law of total probability or total probability theorem.

Doing several events it is known that the probability of all the possibilities should be known. The theorem of total probability is the core foundation of Baye’s theorem. In this article, we have discussed important concepts related to total probability and those are the law of total probability, statement, proof, and some examples

Prerequisite – Random variables, Conditional probability

## Law of Total Probability

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 occurred.

## Total Probability Theorem 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 the addition theorem of probabilities for a union of disjoint events. Using Conditional Probability

```P(B / A) =  P(B ∩ A) / P(A)
```

Or by the multiplication rule,

```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 the 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 Of Theorem Of Total Probability

It is used for the evaluation of the denominator in Bayes’ theorem. Bayes’ Theorem for n set of events is defined as,

Let E1, E2,…, En be a set of events associated with the sample space S, in which all the events E1, E2,…, En have a non-zero probability of occurrence. All the events E1, E2,…, E form a partition of S. Let A be an event from space S for which we have to find probability, then according to Bayes’ theorem,

P(Ei|A) = P(Ei)P(A|Ei) / ∑ P(Ek)P(A|Ek)

for k = 1, 2, 3, …., n

## Example

• We draw two cards from a deck of shuffled cards with replacements. Find the probability of getting the second card a king.

Explanation – 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 – represents 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 the 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 the 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
```

## FAQs on the Law of total probability

Q1. What is the use of total probability?

Law of total probability is used to calculate the probability of an event given any number of related events. Using Baye’s Theorem to update the probability of a hypothesis given new evidence.

Q2. Is total probability always 1?