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The degree of an event’s chances 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.
Multiplication Rule of Probability
The chance of two separate events being multiplied together yields the probability that both events will occur. The probabilities of each independent occurrence must be known in order to use the rule. The multiplication rule states that given these events, the probability that both occur is calculated by averaging the probabilities of each event.
The Multiplication Rule Formula
Using mathematical notation makes it much simpler to explain and apply the multiplication rule.
Events A and B, along with their respective probabilities, are denoted by P(A) and P (B). If A and B happen separately, then:
P(AB)= P(A) P(B)
Even more symbols are used in certain variations of this formula. The intersection symbol . can be used in place of the word “and.” This equation is occasionally used to define separate events. If and only if P(A and B) = P(A) P, then events are independent (B).
For Dependent event multiplication rule
Dependent events are ones in which the probability of one event changes when the likelihood of the other event changes.
P (AB)=P(A) P (B | A)
For Independent event multiplication rule
These events are referred to as independent events if one remains unaltered or unaffected while the other is happening. As a result, even if two events are unrelated to one another, we may still determine their likelihood of occurring. The probability of two connected events is determined by the multiplication rule as follows:
P (A ∩ B) = P (A) P (B | A)
Since A and B are independent, there will be no relationship between them. Furthermore, conditional probability will not be used in this situation. The likelihood of two separate events is therefore given by:
P (A ∩ B)=P (A) . P (B)
Proof of multiplication theory
The probability multiplication theorem tells us that the following is how to represent the conditional probability of an event A when B has happened:
P (A | B) = P (A∩B) . P (B)
Where,
P (B) ≠ 0
P (A∩B) = P (B) × P (A | B) (i)
P (B | A) = P (B ∩ A) . P (A)
Where,
P(A) ≠ 0
P (B ∩ A) = P (A) × P (B | A)
P (A ∩ B) = P (B ∩ A)
We have,
P (A ∩ B) = P (A) × P (B | A) (ii)
From the equation (i) and (ii) we get,
P (A ∩B)= P (B)×P (A | B)=P (A) × P (B | A)
Where
P A≠ 0
P (B) ≠ 0
As a result, the probability multiplication theorem is the above conclusion.
Examples of multiplication rule in probability
Example:
With a standard 6-sided dice, what is the likelihood of getting a 5 and subsequently a 2?
Solution:
Sample space is {1, 2, 3, 4, 5, 6}
Total events is 6
Probability of getting a 5 =1 /6
Probability of getting a 6 =16
Using the probability multiplication rule for independent events,
Pa5 and a2 = 1 / 6.1 / 6=1 / 36.
With a standard 6-sided dice, the probability of receiving both a 5 and a 2 is therefore 1 /36.
Conclusion
In this article we learned that the probability of two events A and B occurring simultaneously is the product of the likelihood of each event given that the first has already happened. The probability Multiplication Theorem states that this is the case. The multiplication rule uses the individual event probabilities to determine the likelihood of many occurrences occurring simultaneously. The likelihood of occurrences A and B occurring together can be calculated using the multiplication rule if events A and B happen separately.
