Multiplication Rule of Probability

Probability is defined as the ratio of the number of favourable outcomes to the total number of possible outcomes of an event. In an experiment with ‘n’ number of outcomes, the number of favourable outcomes can be stated as x. The formula for calculating the probability of an event is as follows.

Probability (Event) = Favourable Outcomes/Total Outcomes = x/n 

Probability determines the likelihood of an event occurring. There are numerous circumstances in which we must predict the outcome of a real-life occurrence. We may be certain or uncertain about an event’s outcome. In these cases, we say the event has a possibility of occurring or not occurring. Probability has a wide range of uses in gaming, business (for making probability-based projections), and artificial intelligence.

The chance of an event can be calculated using the probability formula by dividing the favourable number of possibilities by the total number of possible outcomes. Because the number of favourable outcomes can never exceed the total number of outcomes, the probability of an event occurring can range from 0 to 1. Furthermore, there can’t be a negative number of positive outcomes.

• Properties of Probability

A null set is the probability of an improbable event.

The maximum probability of an occurrence is its sample space (sample space is the total number of possible outcomes)

The probability of any event varies from 0 to 1. (0 can also be a probability).

It is impossible to have a negative probability for an event.

If A and B are mutually exclusive outcomes (two occurrences that cannot occur simultaneously), the probability of either A or B occurring is equal to the probability of A plus the probability of B.

• Multiplication Rule of Probability

When an event is the intersection of two additional events, such as occurrences A and B, the multiplication rule of probability stipulates that both events must occur at the same time. Then P(A and B)=P(A)*P( (B). The set AB denotes the occurrence of both events A and B at the same time, that is, the set in which both events A and B have occurred. The letter AB can be used to represent the event AB.

P(A ∩ B) = P(A) P(B | A).

• Probability Multiplication Rule for Dependent Events

Dependent events are ones in which the outcome of one event has an impact on the outcome of another. The probability of the second event is sometimes influenced by the previous event’s occurrence. P(A ∩ B) = P(A) P(B | A) is derived from the theorem, where A and B are independent events.

• Probability Multiplication Rule for Independent Events

Independent events are defined as those in which the outcome of one event has no bearing on the outcome of another. The probability multiplication method for dependent events can be extended to independent occurrences. P(A ∩ B) = P(A) P(B | A), therefore if the events A and B are independent, P(B | A) = P(B), and the previous theorem becomes P(A ∩ B) = P(A) P(B). That is, the probability of both of things happening at the same time is the product of their probabilities.

• Formula for the Multiplication Rule of Probability

The probability of two events, A and B, occurring simultaneously is equal to the probability of B occurring times the conditional probability of A occurring given that B occurs, according to the multiplication rule of probability.

P(A∩B)=P(B)⋅P(A|B)

Simply multiply both sides of the conditional probability equation by the denominator to obtain the general multiplication rule of probability.

• Proof of Multiplication Rule of probability

The properties of conditional probability are used to calculate the chance of two events, A and B, intersecting.

We know that P(A|B) denotes the conditional probability of event A given that B has occurred, and that P(A|B) = P(A∩B)P(B), where, P(B)≠0. P(A∩B) = P(B)×P(A|B) …….(1)

P(B|A) = P(B∩A)P(A),where, P(A) ≠ 0. P(B∩A) = P(A)×P(B|A)

As, P(A∩B) = P(B∩A), P(A∩B) = P(A)×P(B|A) … …..(2)

From equation (1) & (2), 

P(A∩B) = P(B)×P(A|B) = P(A)×P(B|A), P(A) ≠ 0,P(B) ≠ 0. 

Thus, the obtained result is known as the multiplication rule of probability.

Also, in the case of independent events A and B, P(B|A) = P(B). 

The obtained equation (2) can be altered as, 

P(A∩B) = P(B) × P(A)

• Probability Multiplication Rule for n Events

Here, to get the probability multiplication rule for n events, we must extend the probability multiplication theorem to n events for n events.A1, A2, … , An, 

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

Now for the case of n independent events, the multiplication theorem get reduced to P(A1 ∩ A2 ∩ … ∩ An) = P(A1) P(A2) … P(An).

Conclusion

The multiplication rule is a method for calculating the probability of two events occurring at the same time (this is also one of the AP Statistics formulas). There are two rules for multiplication.P(A ∩ B) = P(A) P(B|A) is the general multiplication rule formula, and P(A and B) = P(A) * P(B) is the particular multiplication rule formula . P(B|A) denotes “the probability of A occurring given that B has occurred.”