experiment. The probability P is a real valued function whose domain is the power set of S and range is the interval [ , ] satisfying the following axioms For any event E, P (E) ≥ P (S) = (iii) If E and F are mutually exclusive events, then P(E ∪ F) = P(E) + P(F). It follows from (iii) that P( φ ) = . To prove this, we take F = φ and note that E and φ are disjoint events.
Therefore, from axiom (iii), we get P (E ∪ φ ) = P (E) + P ( φ ) or P(E) = P(E) + P ( φ ) i.e. P ( φ ) = . Let S be a sample space containing outcomes , ,..., ω ω ω , i.e., S = { ω , ω , ..., ω n } It follows from the axiomatic definition of probability that ≤ P ( ω i ) ≤ for each ω i ∈ S P ( ω ) + P ( ω ) + ... + P ( ω n ) = (iii) For any event A, P(A) = ∑ P( ω i ), ω i ∈ A.
A Note It may be noted that the singleton { ω i } is called elementary event and for notational convenience, we write P( ω i ) for P({ ω i }). For example, in ‘a coin tossing’ experiment we can assign the number to each of the outcomes H and T. i.e. P(H) = and P(T) = ( ) Clearly this assignment satisfies both the conditions i.e., each number is neither less than zero nor greater than and PROBABILITY P(H) + P(T) = + = Therefore, in this case we can say that probability of H = , and probability of T = If we take P(H) = and P(T) = ...
( ) Does this assignment satisfy the conditions of axiomatic approach? Yes, in this case, probability of