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Probability Formulas and Equation

Discussion in 'Study Lounge' started by Royaluni, Dec 1, 2012.

  1. Royaluni

    Royaluni Administrator

    Probability can be defined as the possibility of any occurrence and it is expressed in numerical values which range from 0 to 1. 0 represents impossibility and 1 represent absolute certainty.
    Probability is expressed in ratios like 13/53 mean 13 times a particular event can happen whereas 53 is the total number of happenings. Now say there are 53 cards and 13 is the number of one subtype (X type) of card type.

    Probability postulates
    S denote the sample space of a random experiment
    Oi are the basic outcomes
    A is an event
    For each event A of the sample space S, we assume that a number P (A) is defined and we have the postulates.
    If A is any event in the sample space S
    0<P (A) <1
    Let A be an event in S, and let Oi denote the basic outcomes. Then P (A) =∑ P (Oi) where the notation implies that the summation extends over all the basic outcomes in A.

    Probability Definition

    Classical Definition – the classical definition of probability is the probability of an event will occur. The probability of an event is calculated by dividing the “number of an event happens in sample” by the “total number of occurrence”.

    The probability of an event A is

    P (A) = NA/N

    Where NA is the number of outcomes that satisfy the condition of event A and N is the total number of outcomes in the sample space. The important idea here is that one can develop a probability from fundamental reasoning about the process.

    Probability Rules and Formulas

    Complement Rule
    Let A be an event and its complement. Then the complement rule is:
    =1-P(A)

    The Addition Rule of Probabilities
    Let A and B be two events. The probability of their union is
    P(A U B ) = P ( A ) + P( B ) - P( A ∩ B )

    Conditional Probability
    Let A and B be two events. The conditional probability of event A, given that event B has occurred, is denoted by the symbol P( A|B ) and is found to be:
    P(A/B) = P(A∩B)/P(B)

    The Multiplication Rule of Probabilities
    Let A and B be two events. The probability of their intersection can be derived from conditional probability as
    P( A ∩ B) = P( A|B ) P( B )

    Statistical Independence
    Let A and B be two events. These events are said to be statistically independent if and only if
    P( A / B ) = P( A ) P( B )
    From the multiplication rule it also follows that
    P( A|B ) = P( A ) ( if P( B ) > 0 )
    More generally, the events E1, E2, …., EK are mutually statistically independent if and only if
    P( E1 ∩ E2 ∩ …..∩ EK ) = P( E1 ) P( E2 )…..P( EK )
     

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