Multiplication Rules of Probability

Introduction

The multiplication rules of probability define the proper way to find the relationship of occurrence between dependent and independent events. To find this relationship, the rules and notations of the set rule should be known because conditional probability uses set notations for defining the relationship between two events. Hence, knowledge of the set rule, probability, and basic multiplication rules are required to proceed with solving conditional probability. This multiplication rules of probability study material provide the necessary conditions that define the dependency and independence of a certain set of events to apply proper rules and techniques.

Basic Criteria for Multiplication Rules of Probability

The probability of an event defines the number of times it has occurred considering other given factors. Similar rules need to be followed in addition to the basic multiplication rules when conditional probability events have to be solved. The only condition is that the probability of both events occurring simultaneously is required which means if event A and event B have occurred simultaneously then we need to figure out the probability for this condition.

• The probability of two events, e.g., event A and event B, is given by P(A) and P(B), respectively.
• The probability of every event belongs to the sample space set denoted by S. Here, P(A) and P(B) also belong to the universal sample space S.
• The representation when both the events, i.e., A and B, have occurred is given by using the intersection symbol, i.e., A∩B
• Therefore, the probability that both events have occurred is given by P(A∩B). It is similar to the probability of finding any events; the only thing is that we need to use proper set notations to define intersection or union as per the requirement.
• These terms, i.e., P(A), P(B), and P(A∩B) are used in a suitable relation defined by the multiplication rule that depends on whether the events are dependent or independent.

In general, the multiplication rules provided for probability states that the probability of simultaneous occurrence of two events is equal to the product of the probability of their occurrence after considering their dependency and independence criteria. In this way, a proper probabilistic relationship is formed.

Multiplication Rules of Probability for Dependent Events

If one event is dependent on the other event, then the dependency needs to be considered while evaluating their conditional probability. In this case,

• The probability of event N is considered, i.e., P(N)
• If event M is dependent on event N, the conditional probability, which signifies that event M occurs if event N has occurred, is given by P(M/N), where M/N signifies that M is dependent on the occurrence of event N
• The final multiplication rules of probability state that the product of event N and the conditional probability of event M gives the final probability of these dependent events.
• This is represented by P(M∩N) = P(N)*P(M/N).

Multiplication Rules of Probability for Independent Events

The case of multiplication rules of probability for independent events is one of the simplest cases of probability because there is no specific or hidden relationship between the events. In a dependent event case, there are various hidden clues to determine the dependency of events, unlike the case of independent events. There are no specific multiplication rules for independent event scenarios. In this case,

• If two events, e.g., M and N, are given, their probability must be evaluated first.
• The probability of event M will be P(M), and that of event N will be P(N); this is one of the major steps for finding the final probability value.
• The final probability, i.e., the probability of event M and N occurring simultaneously, is given by P(M∩N)
• According to the multiplication rules given for probability, the final equation to evaluate probability is given by P(M∩N) = P(M)*P(N)

The final probability product gives the fractional value representing the chances of occurring events M and N simultaneously.

Theorem for Multiplication Rules of Probability

The proof or theorem for the establishment of multiplication rules of probability is necessary to be applied in required conditions and cases where it satisfies all the required criteria. For this, the selection of criteria is important as it should not violate the basic multiplication rules and basic probability considerations that are necessary.

There were two ways of calculating the final probability product from the multiplication rules of probability for dependent and independent events.

Let’s take two events, M and N, whose probability is given by P(M) and P(N), respectively.

If two events are dependent, then the probability is given by

• P(M∩N) = P(N)*P(M/N) when M is dependent on N and N is not 0
• P(M∩N) = P(M)*P(N/M) when N is dependent on M and M is not 0

If two events are independent, then the probability is given by

• P(M∩N) = P(M)*P(N)

From the set rule, we know that for independent events

• P(M/N) = P(M)

Therefore, in P(M∩N) = P(N)*P(M/N)

• P(M∩N) = P(M)*P(N)

Conclusion

In a broader view, the dependent events can be converted to independent by using suitable set relations, and the problem can be solved easily. If the conversion of the dependent event into the independent event is not possible, then using the set rules, the conditional probability of the dependent event can be found. This helps to get the final probability of the two events easily. The only necessary thing is to make sure that the hidden dependency conditions and variables are evaluated beforehand.