Multiplication Theorem

The focus of this paper is on the Multiplication theorem which is connected to the occurrence of events whether events are independent or co-dependent. There are two different things such as the Multiplication Theorem for independent events and the multiple theorem of probability. Both are different things, however the concept remains the same, that is the occurrence of events. Elaborative details will be given in the further discussion regarding the concept of Multiplication Theorem and the formula of Multiplication Theorem.

Multiplication theorem

What is the multiplication theorem?

In case A is referred to as an independent event and so as B then the chances for both of them to emerge is the same with the product of their individual chances to appear. This concept is known as the Multiplication Theorem and the multiplication theorem formula is P (A ∩ B)= P (A)*P (B).

For example, if the event A can happen in n1 ways where p has achieved its success then,

B can also happen in 2 ways where q has achieved its success. The combination of both the successes present the overall number of successful events, which is p*q. The total cases that are here in this context are n1*n2. 

Moreover, it can be stated that the conditions of the multiplication theorem depend on the occurrence of two events and the formula is developed as per the changes noticed in the occurrence of the events. It can be different for the events that have the probability to appear together simultaneously and it works slightly different for the independent events.

Formula of the multiplication theorem?

The formula of the multiplication theorem is subjected to be used in the context to measure the probability of two events to appear at the same time simultaneously. The formula also helps in identifying the number of chances here of total probable outcomes. The formula that is mostly used is P (A ∩ B) =P (A)*P (B).

Multiplication theorem of probability

At times when an action occurs such as throwing a ball or lifting up a weight, then there is some sort of probability linked with it. This single concept can have many perspectives in different subjects, however; as per the language of mathematics probability signifies the ratio between desired outcomes and the overall count of total outcomes that can occur. The probability for two events to occur side by side is the same with the product of chances of another item. The only condition that must be applied in this context is that the first event has already emerged. This is what is known as the Multiplication Theorem of probability. At times, when two events occur parallelly, the first one is pointed as A and the second one is pointed as B. therefore, a formula can be stated that in case A and B refer to two different events that are associated with a singular experiment then,

P (A ∩ B) = P (A) P (B/A) condition, P (A) ≠ 0 and one more thing is possible that is, 

P (A ∩ B) = P (B) P (A/B) condition, P (B) ≠ 0.

Multiplication theorem for independent events

The context of the multiplication theorem for independent events is that two events are not subjected to be interconnected as they are different, however; there is a chance that they can appear at the same time. As per this condition, the probability of their individual occurrences can be equally compared to the chances that they have to appear together. The basic formula has been stated in the earlier part of the document which is, P (AB) = P (A) * P (B). 

A mathematical calculation always requires to be proved and that is why in case m1 is the number of cases associated with the emergence of the event tagged as A out of n1 exhaustive and similar sorts of cases. P (A) = m1/n1. 

In case m2 is the number of cases associated with the emergence of the event tagged as B out of n2 exhaustive and similar sorts of cases. P (B) = m2/n2.

Therefore, with the help of the fundamental principle for counting, the total count of the cases associated with the emergence of the event AB is m1m2 on n1n2. Therefore, the combined formula that is generated in this context is, P (AB) = m1m2/n1n2 and P (AB) = P(A) * P(B) is proved.

Conclusion

The assignment is based on the multiplication theorem and the theorem is based on the occurrence of two different events. It has been found out through the formulas that are connected with this theorem that if there are high chances for two events to appear at the similar time then the product of the probability of their individual appearances are equal. It is important to compare the outcome with the total number of probable outcomes to get the best result.