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What is Probability?. American Heritage Dictionary Definition 3: “ Math . A number expressing the likelihood of occurrence of a specific event, such as the ratio of the number of experimental results that would produce the event to the total number of results considered possible.”

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What is probability
What is Probability?

  • American Heritage Dictionary Definition 3: “Math. A number expressing the likelihood of occurrence of a specific event, such as the ratio of the number of experimental results that would produce the event to the total number of results considered possible.”

  • AHD Definition 1 of Likelihood: “The state of being likely or probable; probability.”


From the web
From the web

  • “Probability is the name given to the branch of mathematics that deals with chance and how to predict whether a result is likely or unlikely.”(http://www.learn.co.uk/default.asp?WCI=Unit&WCU=275)

  • “By probability, we generally mean the likelihood of a particular event occurring, given a particular set of circumstances.The probability of an event is generally expressed as a quantitative measurement.”(http://www.maps.jcu.edu.au/hist/stats/quet/quet6.htm)


Compare
Compare:

  • What is time?

  • What is a point?


Probability of an event three perspectives
Probability of an Event: Three Perspectives

  • Classical (“A priori” or “theoretical”)

  • Empirical (“A posteriori” or “Frequentist”)

  • Subjective


Classical probability a priori or theoretical
Classical Probability (“A Priori” or “Theoretical”)

  • Situation: “experiment” or “random process” with n equally likely outcomes.

  • E.g, toss a fair die: Six equally likely outcomes,

  • P(A) = m/n, where A is satisfied by exactly m of the n outcomes

  • E.g., toss a fair die; A = an odd number comes up -> P(A) = 3/6.


Pros and cons of classical probability
Pros and Cons of Classical Probability

  • Conceptually simple for many situations

  • Doesn’t apply when outcomes are not equally likely.

  • Doesn’t apply when there are infinitely many outcomes


Empirical probability a posteriori or frequentist
Empirical Probability (“A Posteriori” or “Frequentist”)

  • P(A) = limn --> ∞(m/n), where n = number of times process performed, m = number of times A is satisfied.

  • E.g., toss a fair die; A = six lands up

  • E.g., toss a die that is suspected of not being fair; A = six lands up.


Pros and cons of empirical probability
Pros and Cons of Empirical Probability “Frequentist”)

  • Covers more cases than classical.

  • Intuitively agrees with classical when classical applies.

  • Repeating the identical experiment an infinite number of times (sometimes even twice) is physically impossible.

  • How large must n be to give a good approximation to the limit?


Subjective probability
Subjective Probability “Frequentist”)

  • A person’s measure of belief that some given event will occur.

  • E.g., P(the stock market will go up tomorrow).

  • Needs to be “coherent” to be workable. (e.g., P(stock market goes up tomorrow) = .6 and P(stock market goes down tomorrow) = .7 are inconsistent.)


Pros and cons of subjective probability
Pros and Cons of Subjective Probability “Frequentist”)

  • Applicable in situations where other definitions are not.

  • Fits intuitive sense of probability.

  • Can be considered to extend classical.

  • Can vary from individual to individual

  • Requires “coherence” conditions; are people always that rational?


Unifying perspective axiomatic model of probability
Unifying Perspective: Axiomatic Model of Probability “Frequentist”)

A function P from events to non-negative numbers satisfying:

  • 0 ≤ P(E) ≤ 1

  • P(S) = 1 (S = certain event; sample space)

  • P(union of mutually exclusive events) = sum of P of individual events