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The degree of an event’s likelihood is referred to as its probability. There must be some probability attached to an event as it happens, such as when a ball is thrown, a card is drawn, etc. Probability in mathematics is defined as the proportion of desired outcomes to all potential outcomes. Given that the first event has already happened, the likelihood that events A and B will occur simultaneously is equal to the product of the probabilities of the two events. The probability Multiplication Theorem refers to this.
The condition between two given events is described by the probability multiplication rule. A∩B represents the occurrences in which two events, A and B, connected to a sample space S, have occurred. In probability, this is often referred to as the multiplication theorem. The likelihood that the two given occurrences will occur concurrently is calculated by multiplying the probabilities of the two given events.
Definition of Multiplication Theorem
The condition between two given events is described by the probability multiplication rule. A∩B represents the occurrences in which two events, A and B, connected to a sample space S, have occurred. In probability, this is often referred to as the multiplication theorem. The likelihood that the two given occurrences will occur concurrently is calculated by multiplying the probabilities of the two given events.
How to Prove the Multiplication Theorem
Using the probability multiplication rule , we can derive
P (A ∩ B) = P (A) × P (B|A); if P(A) ≠ 0
P (A∩B) = P (B) × P (A|B); if P (B) ≠ 0
According to the probability multiplication theorem, the probability of two independent events, X and Y, occurring simultaneously in a random experiment is equal to the product of their probabilities.
P (A∩ B) = P (A) × P (B)
Additionally, the multiplication rule shows us that
P (A∩B) = P (A) × P (B|A)
Given that X and Y are separate events,
P (B|A) = P (B)
Then we obtain,
P (A∩B) =P (A) × P(B)
This indicates that the multiplication theorem is correct.
Formula of Multiplication Theorem
According to the probability multiplication rule, the likelihood that events A and B will occur simultaneously is equal to the likelihood that B will happen multiplied by the likelihood that A will happen if B happens.
P(A∩B)=P(B)P(A|B) is a valid representation of the multiplication rule.
Simply multiplying both sides of the conditional probability equation by the denominator will yield the general multiplication rule of probability.
Probability Multiplication Rule for Dependent Events
Dependent events are ones in which the outcome of one event has an impact on the outcome of the other. Sometimes, the likelihood of the second event depends on whether the first event occurs.
P(A ∩ B) = P(A) P(B | A)
where A and B are independent events, follows from the theorem.
Probability Multiplication Rule for Independent Events
Events are considered to as independent events if the outcome of one event has no bearing on the outcome of another. For independent occurrences, the probability multiplication rule that applies to dependent events can be expanded. If the events A and B are independent, then
P(B | A)= P(B)
and the aforementioned theorem reduces to P(A ∩ B) = P(A) P(B). This implies that the likelihood of both of them happening at once is the sum of their individual probabilities.
Conclusion
In this article, we learned The Multiplication Rule of Probability states that you should multiply the two probabilities to determine the likelihood that two events will occur together. The intersection of the two events is the term used to describe the likelihood of two occurrences happening together. The likelihood of many occurrences occurring simultaneously can be calculated using known probabilities of each event separately thanks to the multiplication rule. The particular and broad multiplication rules are the two variations of this rule.
