Addition and Multiplication Theorems of Probability

Introduction

Consider the following scenario: You participate in a dice game with your friends. You create an exhaustive list of all the potential outcomes in the sample space. When one of your friends rolls the dice, he does not reveal the outcome to you. Rather than that, he states that when the two numbers are added together, they provide an even number. At this point, can you predict the outcome of his coin toss? It would be fascinating to examine how this new knowledge influences the probability of different outcomes. This fundamental assumption underpins the probability multiplication theorem. 

Probability Theorem Based on the Addition and Multiplication Theorems of Probability

As conditional probability teaches us, the likelihood of an event occurring changes when one or more likely occurrences arise. When we know that an event B happened, we focus on B rather than S when estimating the probability of event A happening, given that B occurred.

The following are the most probable outcomes of tossing two dice in the case above:

S = (x, y): x, y equals 1, 2, 3, 4, 5, 6; 

The sample space S has a total of 36 items. Every possible solution has a P(Ei) value of 1/36 in terms of likelihood of occurrence. We have no clue how our friend’s dice roll will turn out. The sum of the integers, on the other hand, is an even number. Consider how this knowledge impacts the probability of the result occurring in the future.

Each of these events have a chance of occurrence of 1 in 18, denoted as P(A |Ei). This example demonstrates how the availability of fresh knowledge may alter the probability of an event happening.

Probability of Occurrence of an Event

This is the first of these theorems:

When two events A and B occur, in the instance of P(A ∩B), it is identical to P(A |B) P(B). 

P(A ∩ B) = P(A) P(B | A), P(A) > 0.

or, P(A ∩ B) = P(B) (A | B), P(B) > 0.

P(B | A) is the conditional probability that event B will occur if event A has already happened. P(A | B) is the conditional probability that event A will occur if event B has previously occurred in this scenario.

We may conclude the following:

P(A | B) = P(A ∩ B) ⁄ P(B).

P(A ∩ B) = P(B) P(A | B).

P(B | A) = P(A ∩ B) ⁄ P(A).

⇒ P(A ∩ B) = P(A) P(B | A).

When two events A and B occur concurrently, a probability theorem demonstrates that the chance of their happening is equal to the product of the probabilities of one event and the conditional probabilities of the other, assuming the first event happened.

The second theorem is as follows: 

n(A ∩ B) ≤ n(A) … (i),

and, n(B) ≤ n(S) … (ii)

Dividing (i) and (ii), we get,

n(A ∩ B) ⁄ n(B) ≤ n(A) ⁄ n(S)

⇒ P(A | B) ≤ P(A).

This is highly intuitive as the probability with a condition applied will always be lesser than that without the condition.

The Multiplication Theorem of Probability

As seen in the example below, the probability multiplication theorem for dependent events may include independent occurrences. 

P(A ∩ B) = P(A) P(B | A). 

Assume that occurrences A and B are unconnected i.e A and B are independent events, then

P(B | A) = P(B). 

The preceding theorem may be abbreviated as follows:

In this situation , P(A ∩ B) = P(A) P(B).

This means that the likelihood of both of these events occurring concurrently is equal to the product of their individual probabilities.

The probability multiplication theorem is extended to include occurrences with n independent events.

We have n independent events A1, A2,…, An

 P(A1 ∩ A2 ∩ … ∩ An) = P(A1) P(A2) … P(An).

The probability multiplication theorem is extended to include n distinct events.

The multiplication theorem is reduced to its simplest form for occurrences of n events.

P(A1 ∩ A2 ∩ … ∩ An) = P(A1) P(A2 | A1) P(A3|A1∩ A2) … × P(An |A1 ∩ A2 ∩ … ∩ An-1)

Conclusion

Working on as many probability questions as possible is the best approach to understanding when to add and multiply. However, if the sentence contains the word “or,” add the probability. Multiply the probability if the sentence contains the word “and.”