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on the space This is usually read as probability of B given A If this conditional probability of B given A is the same as the probability of B, then B and A are said to be independent In the case that the sample space is nite or countably in nite, a probability function can also be de ned by its values on the elementary events { 1 }, { 2 }, , where = { 1 , 2 , }

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In a context proper to probability theory, the properties of a probability measure P : F [0, 1] are imposed under the form of axioms known as the Kolmogorov axioms: K1 For any set E, 0 u P (E ) u 1 (A5)

That is, the probability of an event set is represented by a real number between 0 and 1 K2 P ( ) = 1 (A6)

That is, the probability that some elementary event in the entire sample set will occur is 1 More speci cally, there are no elementary events outside the sample set This is often overlooked in some mistaken probability calculations; if one cannot precisely de ne the whole sample set, then the probability of any subset cannot be de ned either

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K3 Any countable sequence of mutually disjoint events E 1 , E 2 , satis es P (E 1 E 2 ) = P (E 1 ) + P (E 2 ) + (A7)

We have seen how to focus resources on the extension ideas with the biggest potential for brand and business building We will now go on to look at the important but often overlooked issues of execution The next chapter on delivery will highlight the need to build multiple levels of differentiation into your product or service to avoid the mistake of overpromising

Section 101: Public Company Accounting Oversight Board (PCAOB) Membership The board shall consist of five full-time members (two CPAs and three non-CPAs) who are all financially literate No member of the board may be receiving payment or sharing in the profit of any public accounting firm other than retirement benefits or other fixed payments The chair may not have practiced as a CPA within the previous five years Section 103: PCAOB s Duties The board is responsible for: Setting the budget and managing its operations Establishing auditing, quality control, ethics, independence, and other standards relating to the preparation of audit reports for issuers Registering and inspecting accounting firms Investigating irregularities and imposing appropriate sanctions Enforcing compliance with the Act and other laws or standards relating to the preparation and issuance of audit reports Performing other duties as required The board must adopt an audit standard to implement the internal control review required by Section 404 Section 105: PCAOB Investigations Information received or prepared by the PCAOB shall be confidential and privileged as an evidentiary matter (and shall not be subject to civil discovery or other legal process) in any proceeding in any Federal or State court or administrative agency, unless and until presented in connection with a public proceeding or [otherwise] released No sanctions report will be made available to the public unless and until stays pending appeal have been lifted Section 107(d): PCAOB Sanctions The SEC has the right to require the board to carry out additional responsibilities, such as keeping certain records, and it can inspect the board as necessary Section 107(c): Review of Disciplinary Action Taken by the PCAOB The SEC can change, cancel, reduce, or increase sanctions applied by the board Section 108: Accounting Standards The SEC recognizes GAAP and all the principles therein, and any new procedures must adhere to the GAAP principles.

That is, the probability of an event set which is the union of other disjoint subsets is the sum of the probabilities of those subsets This is called -additivity If there is any overlap among the subsets, this relation does not hold A probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satis ed In technical terms, a probability distribution is a probability measure whose domain is the Borel algebra on the reals

(i) From the Kolmogorov axioms one deduces other useful rules for calculating probabilities: P (A B) = P (A) + P (B) P (A B) (A8)

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