Multiplication Theorem

In this article we see some multiplication theorem in probability. The probability of the simultaneous occurrence of two events E1 and E2 is equal to the product of the probability of E1 and the conditional probability of E2 given that E1 has occurred or it is equal to the product of the probability of E2 and conditional probability of E1 given E2(using multiplication theorem).

I like to share this Mean Value Theorem Calculus with you all through my article.

If E1 and E2 are independent, then P(`(E1)/(E2)` )=P(E2) ¬so that P(E1nE2)=P(E1) × P(E2).

Probability: Probability of an event is defined as the ratio between the number outcomes corresponding to event to the total number of outcomes.

probability  =  number of outcomes corresponding to event

total number of outcomes

Conditional Probability for multiplication theorem:

The probability of occurrence of an event A given that an event B has already occurred is conditional probability that event A will occur given that event B has occurred already and is denoted by P(`A/B` ).

• Probability of an event lies between 0 and 1 (0 `<=`P(E ) `<=`1)

• A rule that can be used to verify a conditional probability from unconditional probabilities is:

P(`A/B` ) = `(P(A nn B ))/(P(B))`

Where:

P (`A / B` ) = the (conditional) probability that event A will happen is given that event B has occurred already.

P (A `nn` B) = the (unconditional) probability that event A and event B both occur.

P (B) = the (unconditional) probability that event B occurs.

Multiplication Rule in multiplication theorem:

• The multiplication rule is a result used to verify the probability that two events A and B both occur.

• The multiplication rules are followed from the definition of conditional probability.

• The result is often written as follows, using set notation:P(A `nn`B )=P(`A/B` )P(B)   or   = P(`B/A` )P(A)

where:

P(A) = probability that event A occurs.

P(B) = probability that event B occurs.

P(A `nn`B)= probability that event A and event B occur.

P(`A / B` ) = the conditional probability that event A occur given that event B has occurred already.

P(`B / A` ) = the conditional probability that event B occur given that event A has occurred already.

Algebra is widely used in day to day activities watch out for my forthcoming posts on Alternating Interior Angles and Central Limit Theorem Equation. I am sure they will be helpful.

Independent Events:

Two events are said to be independent if occurrence of one of the events does not influence the occurrence of other.

P(`A/B` )=P(A)            P(`B/A` )=P(B)

For independent events, that is events which have no work on one another, the rule simplifies to:

P (A `nn` B) = P(A)P(B)